It's true these polynomials are not unimodal for n=2 and up, since they all start with coefficient sequence 1 0 1 ... (The reason for this is clear from the definition of descending plane partitions.) For n = 4, 5, 6, they are also non-unimodal in the middle. But the sequence of nonzero coefficients seems to be unimodal for n=7 and above (I've calculated up to n=35). I had observed the non-unimodality of n=4,5,6 several years ago, but had thought it would continue to fail, so it's interesting to me that it seems to be (nearly-)unimodal at n=7 and beyond.
When the ASM is a permutation matrix, the right statistic is a weighted inversion count: weight each inversion pair by the larger number in the pair and then add up all the contributions. (See St000616 on FindStat.) The generating function for this statistic on permutations is an interesting analogue of the q-factorial. (See https://arxiv.org/pdf/1002.3391v2.pdf Corollary 6.)
I also think looking at the Gog or Magog triangles to try and find the right statistic is a good idea, but it's not one of the usual statistics, as far as I can tell. I've thought about this question quite a lot and would be very interested if anyone is able to make progress on finding this statistic on ASM or TSSCPP.