At this point, considering all you have done, I can think of 2 things.
1, the cooling fins of the head and cylinder may be partially blocked by accumulative debris or mouse activity.
Remove the flywheel housing and look into the sheet metal shrouded air duct surrounding the head and cylinder for signs of grass or other blockage.
2, the gas cap is still blocked inside. Try loosening the cap (don't lose the cap) and see if that solves the problem.
If yes, buy a new cap or try cleaning this one out again. You have to look on the underside of the cap for a hole there as well as the hole outside.

also make sure none of your fuel lines are old/cracked (replace)and that your new fuel filter is specifically the correct one for your engine. Some aftermarket filters have different flow rates...

All fuel lines are in good condition. The fuel filter is the correct part number. Still won't run right.

I am not recommending you do this but what I would do when engine starts to stumble is remove the gas cap, wipe off filler neck, place my lips against and blow, see if this makes engine pick up. NOTE: I would release pressure gradually to prevent gas from blowing back and NOT inhale.

Sounds like classic fuel starvation.

Walt Conner