Guess you're suggesting a chemical stripper. Which should I use, since it
would be working on both layers, which are of different paint?

The top layer would be easy to remove mechanically, but messy. Well, no
paint is easy to remove :-) Bottom (I suspect oil) is very well attached to
the wood.

RichK

"RickH"

If you can get it off with a fingernail then stripping it should be
easy, the new strippers are pretty gentle on your lungs and hands
compared to years ago. Then you could just prime/paint onto the good
layer. No, primer wont penetrate it and somehow re-attach the layer
below.