You did not obtained any logical contradiction.
Logical contradiction by definition is a situation where two statements cannot be true at the same time. Example of logical contradiction would be I am in Amsterdam, and I am in Utrecht at the same time. Since it is impossible both first part of the statement and second part of the statement is true we arrived at a logical contradiction.
In your case your statements are; Big Mac Index is a measure of whether exchange rate is overvalued, and Big Mac Index is based on value of an item that also depends on an exchange rate. Both of these things can be true at the same time.
Consider simpler example, suppose we want to measure if building in Utrecht is taller than expected. A simple measure could be to calculate measure such as follow;
$$T= T_i - E[T]$$
Where $T$ is the measure, $T_i$ is height of building $i$, $E[T]$ is the expected height of the building in Utrecht.
In turn the expectation operator can be written as; $E[T] = \sum_j^n p_jT_j$ where $p$ is probability of selecting some building $j$ and $T$ is the height of building $j$. Next $j$ is the set all buildings in Utrecht that includes the building $i$. Hence we have here also measure that determines whether building $i$, is too tall based on comparing building $i$ to expectation that is again based partially on building $i$ since the value $E[T]$ depends on inclusion of building $i$.
Yet is there any contradiction? Statement building $i$ is taller then expected if $T>0$, and statement building $i$ is included in the set $j$ used to calculate $E[T]$ are completely logically consistent because both statements can be true at the same time.
Similarly one could simply devise a measure of whether currency is overvalued purely by comparing current exchange to its average which would include current exchange rate. It might not be good measure to make investment decisions on but it is not logically inconsistent measure.
In case of Big Mac Index it is based on assumption of law of one price so the Big Mac Index is given by;
$$BM = P_x/P_y$$
where $P$ is price of Big Mac in $x$ and $y$ respectively. Then suppose these prices have some relationship to exchange rate, so $P(S)$ where $S$ is an exchange rate.
You determine whether exchange rate is over or undervalued based on comparison;
$$V=\frac{BM - S}{S} =\frac{P_x(S)/P_y(S) - S}{S} $$
Where $V>0$ means its overvalued, $V<0$ it is undervalued. Now the claim that for some particular country pair;
$$\frac{P_x(S)/P_y(S) - S}{S}> 0 $$
i.e. that the currency is overvalued by this measure, is completely consistent with claim that BMC is a function, (i.e. depends on) of $S$.
Both claims that $V>0$ and $BM$ is composite function of $S$ can be true at the same time. There is no logical contradiction.
