This is a rather complicated question.
I begin with a definition of balanced growth path.
I report, for instance, the definition by Acemoglu, Introduction to Modern Economic Growth, (Princeton University Press, 2009):
Throughout the book, balanced growth refers to
an allocation where output grows at a constant rate, and
capital-output ratio, the interest rate and factor shares remain
constant. (Clearly three of these four features imply the fourth.) (Acemoglu, p. 57, emphasis mine)
Therefore, a constant rate of output growth is a central feature, and, if capital-output is to be constant, also capital must grow at the same constant rate. $^1$
In the Solow model, there is a considerable difference between the case in which the technological progress is introduced through the Harrod neutral (or labor augmenting) technological progress or it is introduced via the Hicks neutral technological progress.
These differences between the two models lead to different conclusions with regard to our problem about the balanced growth path: it can be proved that the Solow model with labor augmenting technological progress is compatible with a steady state/balanced growth path in which the per capita variables grow at a constant rate (equal to the constant growth rate of technology), whereas, if technological progress is introduced through the Hicks neutral form, the model cannot exhibit a steady state with constant growth rates.
Actually, a stronger result holds: technological progress must take the labor-augmenting form in order for the model to have a steady state with constant growth rates. That is, any other form of technological progress is not compatible with a steady state balanced growth.
Moreover, in the case of Hicks neutral technological progress, it is not possible to derive the exact rate of growth of the variables: they grow at a rate depending on the rate of growth of technology, but the specific rate of growth cannot be assessed without knowledge of the specific form of the production function.
Only Harrod neutral technological progress can lead to balanced growth
As we said above, technical progress must have a Harrod neutral representation for the model to have a steady state with a constant growth rate.
Solow himself addressed this question in his book Economic Growth: an Exposition and in Siena Lectures on Endogenous Growth$^2$, pointing out that economic theory is interested in exponential steady states, that is in steady states with constant rates of growth. He writes:
Thus, the devotion of this kind of theory to labor-augmenting
technological progress exactly corresponds to our interest in
exponential steady states. If we were to lose interest in exponential
steady states, then there would be no need to have this assumption
about the nature of technological progress. […]Unless $F $ [the
production function] were this particular form there would never an
exponential steady state. The moral of this part of the story is that
labor augmenting technological progress is not a special assumption
which is needed for this kind of theory to work out; it is a special
assumption which is needed so that we poor people can talk about
exponential steady states. (Solow, Siena Lectures, cit. p. 7, emphasis mine).
In his book and in his Lectures, Solow gives a proof of this fact.
Barro and Sala-i-Martin$^3$ give a proof along similar lines as Solow's.
This result is also known as Uzawa's Theorem:
The theorem [Uzawa's]shows that constant growth of output , capital
and consumption combined with constant returns to scale implies that
the aggregate production function must have a representation with
Harrod neutral (purely labor-augmenting) technological
progress.(Acemoglu, cit. p. 60)
I give a sketch of the proof, following Barro and Sala-i-Martin. For the detailed proofs I refer to the books by Solow, Barro and Sala-i-Martin and Acemoglu.
Sketch of the proof Let us assume a production function that includes both labor-augmenting and capital-augmenting technological progress:
$$Y=F[K \cdot B(t), L\cdot A(t)] \qquad (1)$$
where $B(t)=A(t)$ implies that the technological progress is Hicks neutral.
We assume that $A(t)$ grows at a constant rate $x\geq 0$ and $B(t)$ at a constant rate $z\geq0$, in particular $A(t)=e^{xt}$ and $B(t)=e^{zt}$.
The population , $L$ grows at the constant rate $n$: $L= e^{nt}$.
Dividing both sides $(1)$ by $K$ we have:
$${Y\over K} = e^{zt}\cdot \left (F \left[ 1, \frac {L\cdot A(t)}{K \cdot B(t)} \right ] \right) = e^{zt}\cdot \phi [{L\over K}\cdot e^{(x-z)t}] \qquad (2)$$
where $\phi (\cdot)\equiv F \left[ 1, \frac {L\cdot A(t)}{K \cdot B(t) }\right ]$.
If we were in a balanced growth steady state, the growth rate of $K$ will be constant, say $\gamma _K^*$. The expression $(2)$ for $Y/K$ can be written as:
$${Y\over K}= e^{zt}\cdot \phi [e^{(n+x-z-\gamma _K^*)t}]\qquad (3). $$
In steady state, $\frac{\dot K}{K}$ also equals the constant $\gamma _K^*$$^4$, and, hence, $Y\over K$ must be constant. That is, the right side of $(3)$ must be constant.
There are two ways to get the right side of $(3)$ constant:
Case 1). $z=0$, and $\gamma _K^*= n+x$. That is, technological progress is purely labor augmenting and the rate of growth of capital equals $n+x$ (with $z=0$ we fall back into the usual analysis of labour augmenting technical progress).
Case 2) The production function is a Cobb Douglas.
I omit the technical proof. Just observe that the result is not surprising as, in the Cobb Douglas case, the technical progress can always be thought as labor-augmenting: actually, in a Cobb Douglas the three types of technological progress, labor augmenting, capital augmenting and Hicks neutral are equivalent.
$\Box$
The rate of growth of output with Hicks neutral technology
As we saw, with Hicks neutral technological progress, a balanced growth with constant growth rates doesn't exist: in particular the output doesn't grow at a constant rate.
Actually, we can't even calculate exactly the rate of growth of output, as it requires the knowledge of the specific form of the production function: output grows at a rate that it is linked to the rate of growth of technology, but we don't know the exact rate.
The problem is linked to the fact that with Hicks neutral technology, if technology $A(t)$ grows, say at a constant rate, the steady state shifts, and there isn't a 'state of rest' of the model on a single steady state path with constant $k$, but a continuous shift of the steady state, and therefore a sustained growth of the capital-labor ratio $k$.$^5$
This can be shown through a usual graph of the Solow model, in which the production function is assumed to exhibit Hicks neutral technological progress(as usual, with constant returns to scale), that is
$$Y = A(t) F(K(t),L(t)) = A(t) L(t) f(k(t)),\qquad (4)$$
where the variables have the usual meaning.
In the picture, $A_0 < A_1 < A_2 < A_3 $.
From the graph, representing the fundamental equation of motion, we can see that if $A(t)$ grows, the steady state shifts continuously upward: there isn't one constant steady state value of $k$, but a different value of $k$ for each value of $A(t)$.
We can see this also referring to the fundamental equation which, with Hicks neutral technology, becomes:
$$\dot k = s A(t) f(k(t)- (n+d) k(t)\qquad (5)$$
If we set $\dot k=0$, to obtain the steady state value of $k$, $k^*$, we obtain:
$$s A(t) f(k(t))- (n+d) k(t)=0\qquad (6)$$.
Because of the presence of $A(t)$ in the equation, we see that $k^*$ is not unique for all $t$, but depends on $A(t)$, that in turn depends on time: $k^*= k^*(A (t))$.
So, in time, $k^*(A(t))$ increases as long as $A(t)$ increases, without limit, according to the value of $A(t)$. That is $\dot k^* \neq0$
$$\;$$
We can then calculate the rate of growth of $Y$, $g_Y$ (I omit $t$ for simplicity):
$$Y = A F(K,L) = A L f(k^*) \implies g_Y = g_A + g_L + \frac{f'(k^*)}{f(k^*)} \dot k^* \qquad (7)$$
where $g_A$ and $g_L$ are respectively the rate of growth of technology and the rate of growth of labor.
As during the dynamical process $k^*$ increases, $\dot k^*\neq0$, and we cannot set $\dot k^*=0$ in equation $(7)$ to obtain a constant growth rate of $Y$.
Therefore, the dynamics of $Y$ in time, according to $(7)$, depends on the term $\frac{f'(k^*)}{f(k^*)}$, and it couldn't be exactly determined unless we know the specific form of $f(k)$.
$^1$ Many authors simply define a balanced growth as a situation in which the relevant variables of the model grow at a constant rate, see for example Romer, Advanced Macroeconomics, Mac Graw Hill (2012), p. 18, Dornbush, Fisher, Macroeconomics, McGraw Hill,13°ed.(2017) or Barro, Sala-i-Martin, Economic Growth, MIT Press, (2004) p. 34, footnote 1.
$^2$ Solow, Robert M., Economic Growth: an Exposition, (1970),Clarendon Press, and Solow, Robert M., Siena Lectures on Endogenous Growth,(1992), Siena Department of Economic Policy.
$^3$ Barro, Sala-i-Martin, Economic Growth, cit., p.78.
$^4$ The rate of growth of $K$ is given by $\frac {\dot K}{K}=s(Y/K) - d$.
$^5$ Solow in his seminal paper of 1956, A Contribution to the Theory of Economic Growth, The Quarterly Journal of Economics, Vol. 70, No. 1 (Feb., 1956), introduces technological progress in the Hicks neutral form. He writes: " An especially easy kind of technological change is that which simply multiplies the production function by an increasing scale factor. Thus we alter (2)
to read
(13) $Y = A(t)F(K,L)$.
The isoquant map remains unchanged but the output number attached
to each isoquant is multiplied by $A (t)$. The way in which the (now
ever-changing) equilibrium capital-labor ratio is affected can be seen
on a diagram like Figure I [ the usual graph of the model, my note] by "blowing up" the function sF(r,1)[."p. 85. Then Solow discusses the Cobb-Douglas case.
