I have just had the same problem.
The old gasket glues the step ring to the case and movement so this is what I did.
I pulled the crown out (split stem) and then got a small steel rod with a small curve on the end, I pushed it down the case tube and gently levered the step ring up from outside, it was solid at first, I found putting the watch face up on a warm radiator for 5 minutes helped ease things up a little. Take it slowly and gently.
My brass ring had lost its black in a few places so I brushed it with a fibre pen and then dropped it into Brass black which turned it back to black almost instantly. The finish it just like factory now. If you have dings on your step ring use wet and dry on a very flat surface to get them out, sanding in a circular fashion is best, clean it furiously with a fibre pen and finish with panel wipe, then drop it in brass black.
The gooey gasket had got onto the movement and even on the date wheel stopping it working correctly. It cleans off with gentle perseverance.
The gaskets on your watch look drier than mine so a tiny bit of silicone free lube might help on the outer edge of the ring furthest away from the dial, I would be very careful though.
Good luck and show is when you finish please.
Here’s mine which I did a couple of weeks ago.
