At the risk of devolving into a purely theoretical, impractical exercise (if it's not already too late (which it is)), here goes nothing ;)
There are two cases...

If pseudorandom generation is impossible, then we can tell (by sampling its output) how much true randomness any algorithm uses (call this the true entropy). In this case, the Mersenne Twister is nowhere near big enough. The MT has 2^19937 configurations, so a single MT has at most 19937 bits of entropy. This is nowhere near the 1.5 million bits required to sample the space in question. There would be a polynomialtime algorithm that would be able to tell (by sampling its output) whether or not your algorithm was using MT.

On the other hand, if the MT is really pseudorandom in the strong sense of my previous comment, then we can talk about not only its true entropy but also its computational entropy, that is, the amount of entropy it can "fool" all polynomialtime algorithms into thinking it uses.
From what I recall, if pseudorandom generation turns out to be possible in this strong sense, it is quite reasonable for a function's computational entropy to be much higher (say, by a squared factor) than its true entropy. In this case, MT could be sufficient to sample the desired space.
Essentially, if pseudorandom generation is possible, then bits from the pseudorandom generator are
completely interchangable with truly random bits in the polynomialtime realm. If there is ever a case where it made a (statistically) significant difference in an algorithms output, then already that gives you a distinguishing algorithm that contradicts the definition of the pseudorandom generator! Neat, huh?