You must have read my mind - I was going to suggest a longer barrel, but thought it might be taken as nitpicking
Now that you mention it!!!....
You could do well, I think, with a 24" barrel which is a standard RPK size. Give it a longer piston, like a PSL/Dragunov piston length, and give it a longer gas tube. That would increase velocity a considerable amount, something like around 200-400 FPS, and that would no doubt help.
You could use any 7.62x54r sized barrel, which means any Mosin barrel rebuild option that is out there. Chamber it up and lathe it to the right diameters, "done".
The problem - obviously - is that you have is that the 7.62x39 just doesn't have that much power, and you've reverse-engineered a recoil-operated system into a gas-operated system. That in itself shouldn't be too great a concern, considering the FN Minimi, and the Stoner 63, and the fact that there is a 5.56 version of the MG-42 which was a Spanish design IIRC.
Do you know the muzzle velocity?
I wonder if you can benefit from a muzzle booster, like an AKSU (aka Krinkov) style booster? Would be worth a try, that expansion chamber was designed to boost velocity on the AK-74 based design, so in theory it would help your design as well.
I also wonder if you have the "hammer hump" problem? That seriously robs my .308 bullpup of power to the point that it doesn't always reliably feed. Grind the hump and try it- you never know.
Also, what kind of piston do you have? Home made? Original factory? Reason I ask is that if the piston bleeds enough gas by being narrow enough to make a poor seal, it will let enough gas bleed-off to affect functionality. I had an AK-74 malfunction due to a piston I made being too narrow diameter. Possibly a longer piston head would help, I don't know.
Yes I've put a great deal of thought into this before!!!
I really do want to build something similar to yours.