If you have a shop press that's rated for at least a few tons, you can try reshaping the bottom of the bowl so that it is concave, instead of convex, from how it is right now. To do this right, you'll probably need to heat up the bowl so that it's easier to work with, but the heat from a torch might damage the paint. Even a heat gun might damage the paint, and bending the bowl into a different shape could also destroy the paint.
Then again, I'm going to have to assume you put the ring depression around the hole and that didn't damage the paint, so you might be good to go. Also, if you put in that ring, you probably have the size of shop press to make this happen.
What you'll do is to find or make another heavy bowl or depression for this to sit in while you apply pressure from the top. You might want to use a large piece of rubber or a sandbag with a piece of steel on top to apply even pressure across the dome. The idea is to get the dome to pop out under the bowl or to press it into a different shape with a depression, instead of the raised bit you have. If it's only a 3-5mm, this shouldn't be too difficult, but if the dome does do a "tin can" kind of popping out the bottom, you might have all the pressure of your press released all at once, so strapping the bowl and other press items down would be a good idea.
To be honest, I haven't done anything like this myself, so I'd check to see if there are any tutorials or videos that might have a better way to make this attempt.