It sounds like you're fighting rust and corrosion that may be making it hard for the large o-ring to move along with the housing. You likely won't get the assembly to move out any farther than you've got it now by just using the engagement lever to push it, it's probably at the end of it's travel. Although the pinion/engagement lever was initially difficult to move, there is nothing in the engagement mechanism that's preventing the pinion assembly from coming out. It'll just take a lot more soaking, very careful prying to avoid breaking the mounting ears, and patience. If you haven't already, you'll also need to take the governor off to give you the working room needed to slide the assembly all the way out.
Could it be parts coming apart at the end of the shaft, hanging up on the back side of the flywheel? Try rotating the flywheel slightly.
I had the same problem recently. I took apart the pinion as much as I could ( pull out the shifter lever and shift fork). Now you can grab the pinion case with vice grips and move it up and down while you continue with lubricant. If yours was stuck like mine, this won't get it out.
Thread two bolts into the front of the pinion ( you remove the front cover) and thread the bolts through a piece of chain. Now attach a come along to the front track and the chain. Carefully winch the pinion out while turning it with the aforementioned vice grips.
Sounds crazy - but it was the only way mine would come out - on two occasions!
Don't release the pinion latches, it's not helpful and it makes the whole assembly longer so it's hard to get it to come by the governor once you get it out of the hole.
When I removed mine about 5yrs back I remember I had to remove one of the cap screws on the inspection cover behind the clutch mech. Could this be an issue with your problem?
I removed that cap screw on the inspection cover before I tried it, it has to come a long way before it gets to that, you could probably slide a dime in the separation is all the further it is out --- I will try soaking it more and jiggling it.
I appreciate all your help on this, frustrating dang thing.
Jogging my Memory ccjersey. Your right. Releasing the fingers does make it longer. Keep soaking and wiggling. Dandy Dave!