"I am not given wealth $w$ although I suppose I could assume any firm who
is purchasing has some budget."
No. This is exactly where the fundamental microeconomic theory of the firm differs from the microeconomic Consumer Theory: the firm is not constrained by a budget. The reason is that this fundamental theory deals most and foremost with the "long-term" view, or even better, with the "planning view". So we assume that the amount necessary to cover expenses, will come from sales, since the firm won't enter production at a loss (remember also, this is a deterministic set-up, there is no uncertainty). Working capital considerations (the fact that usually first you have to actually pay expenses and then to actually collect revenues), does not enter the long-term view, justifiably, it is a short-phenomenon. Also, in the long-term or planning approach, there are no fixed costs, all factors are variable.
Now, the "cost-minimization" approach to solve the firm's optimization problem, is an alternative behavioral assumption to the profit-maximizing setup, and it is very relevant in many real-world cases: public utilities that exist mainly to satisfy demand, and their motive is not to maximize profits -rather they want to minimize cost for the given level output, as determined by demand, in the context of efficient use of the always scarce resources.
But also, the case of a price-taking firm that is too small compared to its market, is closer to a cost-minimizing behavior rather than profit maximizing, since the firm has not really control over its production (except downward by direct decision).
In both of the above cases, an exogenous variable appears: the level of output itself. So we solve the problem by treating the level of output as a "constant" or better, we solve it for any given level of output, and the solution we obtain has the level of output as one of its components.
So
$$\min_{K,L} C\equiv rK + wL \\
s.t. F(K,L) = \bar Q$$
with the Lagrangean
$$\Lambda = rK + wL +\lambda[\bar Q - F(K,L)]$$
The first order conditions are
$$r = \lambda F_K,\;\;\; w=\lambda F_L \tag{1}$$
which gives, at the optimum,
$$rK + wL = C = \lambda\big(F_KK + F_L L) \tag{2}$$
Now assume that the production function is homogeneous of some degree $h$ (not necessarily homogeneous of degree one, i.e. exhibiting "constant returns to scale", but homogeneous -and yours is, of degree $h=1/2$.). From Euler's theorem for homogeneous functions of degree $h$ we have that
$$F_KK + F_L L = hF(K,L) = h\bar Q \tag{3}$$
the last equality holding given the constraint of the initial problem. Inserting $(3)$ into $(2)$ we obtain
$$C = \lambda h \bar Q$$
The multiplier $\lambda$ is optimal marginal Cost, denote it $C'(\bar Q)$, so we arrive at
$$C = C'(\bar Q)\cdot (h\bar Q) \implies C'(\bar Q) + [(-1/h\bar Q)]\cdot C =0$$
This is a simple homogeneous differential equation with solution
$$C = A\cdot \exp\left\{-\int(-1/h\bar Q) {\rm d}\bar Q \right\} = A\cdot \exp\left\{(1/h)\ln \bar Q\right\}$$
$$\implies C^* = A\cdot (\bar Q)^{1/h} \tag{4}$$
for some constant $A >0$. To complete the solution, we need to express the object of interest, $C^*$, in terms of the exogenous entities: $r,w,\bar Q$.
To do that derive the optimal marginal cost (which is equal to the multiplier)
$$(4) \implies \lambda^* = (1/h)A(\bar Q)^{1/h-1} \tag{5}$$
Inserting $(5)$ into the first-order conditions we have
$$r = (1/h)A(\bar Q)^{1/h-1} F_K,\;\;\; w=(1/h)A(\bar Q)^{1/h-1} F_L \tag{6}$$
It is time to use the specific functional form of the production function
$$F(K,L) = K^{1/2} + L^{1/2} \implies, F_K = \frac 12 K^{-1/2},\;\; F_L = \frac 12 L^{-1/2} \tag{7}$$
Inerting $(7)$ into $(6)$ together with $h=1/2$ we obtain, after manipulation,
$$rK = \frac {A^2}{r}(\bar Q)^2,\;\; wL = \frac {A^2}{w}(\bar Q)^2 \tag{8}$$
Sum the two to obtain an alternative expression for the Cost function
$$rK+wL = C^* = A(\bar Q)^2\cdot \left[\frac Ar + \frac Aw\right] \tag{9}$$
But inserting $h=1/2$ into $(4)$, we also have that
$$C^* = A(\bar Q)^2 \tag{10}$$
So
$$ (9),(10) \implies A(\bar Q)^2\cdot \left[\frac Ar + \frac Aw\right] = A(\bar Q)^2$$
$$\implies \frac Ar + \frac Aw = 1 \implies A = \frac {wr}{w+r} \tag {11}$$
Inserting $(11)$ into $(4)$ we conclude obtaining
$$C^* = \frac {wr}{w+r}\cdot (\bar Q)^2 \tag{12}$$
Three things:
A) Verify that the second-order-conditions hold for all this to indeed lead to the optimal cost-function.
B) Solve the unconstrained profit-maximization problem with the same production function, normalizing the price of output to $p=1$ (i.e. treating the exogenous prices, $w,r$ as expressed in real terms), to verify that it will lead to a cost level that it is consistent with $(12)$.
C) If you are interested in the theory of the firm under a budget constraint, a related paper is
Lee, H., & Chambers, R. G. (1986). Expenditure constraints and profit maximization in US agriculture. American Journal of Agricultural Economics, 68(4), 857-865.
