I was able to run a 3 cell type parallel test with the LG HE4 cells (in 5p configuration). On discharge, the LG pack provided up to 43.2% of the current, with the Hyundai pack providing up to 53.7% of the load. That puts this 5p LG pack a bit undersized, since it can only provide 100A max (20A/cell) but would source ~130A at that proportion under full throttle. Increasing to 7p should be sufficient, though further tests are needed to see how the proportion scales with pack size (might be a vicious cycle of larger pack => larger load proportion => more current per cell...)
Charging is a bigger problem, the LG pack took up to 40.2% of the charge current, which would give a worst case of 50.25A for the pack, or 10.05A/cell, more than twice the rated max charging current of 4A. To get that down to 4A the pack size would have to be increased to at least 13p, or 11.5kWH nominal.
Hate to say it, but common 18650 or similar form factor cells just aren't cut out for EV use. The last hope would be Sony VTC5As which I've got some of (2nd hand), which are specced at 2.5AH capacity, 6A max charging current, and 35A max discharge current (for short duration). These should be workable in a ~9p pack or ~8kWH nominal, which is still a pretty big pack and will set you back $3520 from China, plus a set of Hybrid cells for $900...
Or you can just wing it with smaller packs and hope they don't catch fire (I do not condone this).
To summarize, here are the estimated minimum pack sizes based on the limited empirical data I've taken so far: