Ruling out Method X is generally much harder than Method X itself, so one shouldn't necessarily expect that for any method that won't work a "certificate" can be found establishing its insufficiency.  With that said:

There are plenty of environments (rings) where the algebraic equation in FLT makes sense, but has solutions.  If the ingredients of Method X apply in one such environment, they aren't using specific enough properties of the integers to prove FLT.  It is well known that if Method X = congruences (reduction modulo $t$ for different values of $t$), then this argument works in rings of $p$-adic numbers where FLT has solutions.   Similarly, if Method X = inequalities, then it would have to rule out the positive real solutions of $x^n + y^n = z^n$.  

Another possibility, requiring much more knowledge, would be to use the close relations between FLT and elliptic curves (Frey, Wiles, etc) to project method X onto the canvas of Wiles' proof and see how much of it can be understood and delimited in those terms.  It could be that X is in effect trying to construct nontrivial objects of a certain kind (esp. cohomology classes) and one can see in light of Wiles methods that the relevant classes are zero, necessary field extensions or coverings are trivial, important obstructions are not killed, etc.  In this approach we work in environments where FLT is true, but try to subsume Method X within existing approaches, and show that it only attains a part of what is needed.

Another idea is "reverse mathematics", to represent Method X as formal derivations in some weak system Y of arithmetic, and show that FLT is of higher proof theoretic strength (because it implies all the theorems of an even stronger system Z).  This is unlikely because FLT is too specialized a statement and the proof-theoretic calibrations of strength only work for reasonably generic theorems with a lot of parameters that can be varied.