As I recall, the only thing holding that shaft in (with the lock plate removed) is the seal that you see from the outside. If there is any play in the shaft, I believe you can just bump the seal out by pushing the shaft in and then quickly tugging on it against that seal (and repeat a few times). You'll realize what's going on as soon as that seal starts to move.
I assume you're going to replace the seal, so you could also just remove the seal destructively. Just be careful not to damage anything but the seal. It's a pretty narrow seal.
There is probably a way to rig a puller on the end of that shaft, or pry from the inside, but I tend to shy away from putting pressure on a cast aluminum case. These cases are pretty thin.