32xx stems should be removed and replaced in the time setting position, same as 31xx. It is usually necessary to press the stem release button in when replacing the stem.
You didn't say if
- you have dial and hands on already
- you are familiar with how to remove and replace a train bridge
In any case, 32xx movements have a new part that interacts with the stem called the intermediate winding gear. It has a spring washer on top of it (bottom of movement side) that pushes the gear down into a stem detent when the detent is positioned under the gear post. This spring washer on rep movements is cheap and flimsy, causing the gear post to get stuck sometimes. You can see the position of this gear on the backside of the movement (see pics below)
In the first pic, the top red arrow points to the top of the gear post. If it is depressed down into the train bridge hole, that may be what is preventing your stem from going in. In this scenario, the problem is not related to the keyless works.
Some more info about this rep winding gear part in the post below. Best option for a fix is just to buy a gen part.
Post #2682
Looking at my 126610 and my gen GMT II, I see a similar cyclops distortion on the gen. It’s not something I ever noticed. Yikes that's going to be hard to unsee, sorry bro. You might have saved us all a bunch of money though. I'm pretty easy when it comes to living with flaws - my gen 16710...
forum.replica-watch.info
If your winding gear spring appears to be functioning correctly (see next post for how to test), then of course the keyless works should be disassembled then reassembled. A 3235 isn't the best movement to learn how to do this, especially since you can't even get a copy of the service manual (it's digital and protected). You could figure it out with a 3135 service manual and by taking pics before and after every part removal.