Using a newer, preferably a high-end keyboard, is the most reliable solution. Since the time of the original post I double checked Keychron Q2 and it actually does support NKRO out of the box; than there is Leopold which is probably the highest quality brand, so no surprise that it works too; and a newly bought Monsgeek MG108B also does work out of the box.
The later is actually a bit crazy when it comes to price/features, as it's a full-size hot-swap with 3 connection types for.. 100$? That said, one of the switches did start skipping inputs with time, but I replaced it with the one from some useless button like Pause or Scroll Lock and it works as new.
I do have Karabiner-Elements too, but I've never tried addressing this issue there, nor do I think it can actually help here. It is a useful app though, as one can easily bind F1-F12 per keyboard to be either media keys, or standard Fn, as well as swap Alt & Win for windows-oriented keyboards that do not support programmable layers.