I had this problem with one I modded myself because the new screen was just slightly thicker than the old, pressing on the outer shell and causing the dpad to need more pressure. The solution that worked for me is cutting a small ring of cardstock and putting it between the dpad and the membrane. It increases sensitivity by a lot, but does come with the side effect that you'll be able to press every direction down at once if you press on the center.

Could it be membrane firmness is the issue, and not necessarily conductivity?

It could be that the d-pad traces on the circuit board have worn down over time and now the conductive material on the membrane isn’t able to make a good connection. You could try and check the traces with the continuity setting on a multimeter if you have one. Maybe very carefully going over the traces with conductive paint could help?, but I have never done that for a d-pad before and don’t know if it would work well or hold up over time.

I've never reshelled my GBA but having installed multiple aftermarket shells on my PSP, I can tell you that d-pad issues are very common because of bad measurements and quality. So maybe the problem is the new shell itself.

So I had this problem on my modded GBA, my solution was to ever so slightly loosen the screws on the dpad side of the shell. Turned out I was over tightening them