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So switching to edge cube positions as cosets (instead of corner cube positions and twist) made a huge impact on the calculation, but more on that later. So my results for up to 16q* for positions at exactly that distance are:

 d (mod M + inv)*     positions
-- -------------- ----------------
 0              1                1
 1              1               12
 2              5              114
 3             17             1068
 4            135            10011
 5           1065            93840
 6           9650           878880
 7          88036          8221632
 8         817224         76843595
 9        7576845        717789576
10       70551288       6701836858
11      657234617      62549615248
12     6127729821     583570100997
13    57102780138    5442351625028
14   532228377080   50729620202582
15  4955060840390  472495678811004
16 46080486036498 4393570406220123

^* The (mod M + inv) column means symmetry reduced postions but only considering the edge cube positions. I just included it because I had those numbers. It is not comparable to earlier calculations because it may include duplicate positions of the full cube and is dependent on what cosets are used in the calculation.