Let's break this down. You can't fit something larger than the bag in, so there are two options - (1) pack it out with something light to keep the small cubes in place, or (2) get slightly larger cubes.
I'd start off with packing out (that is AROUND) the cubes with something light (either some stuff you actually use, or a spacing structure like polystyrene). I wonder whether "outside the box" solutions like an inflatable pillow or two (which you could easily inflate to the right size) wouldn't be perfect here.
If that doesn't work, it might be wise to then buy some cheaper cubes of a larger size to test the fit; worst case scenario would be putting down $$$ on a third size of cubes that doesn't fit. So possibly look for some generic cubes from China to proof of concept the fit first.