I have the same problem trying to remove the entire drain from my mobile home shower stall.
It is the same drain as I've seen in pictures above. The screen does not remove separately. It is welded or something together to the outer ring that also is part of the stem with threads that fits down into the hole. (Sorry, I don't know all the technical terms.)
Also, I have tried to remove screen first because of corrosion Nope, can not do. Went under home and seen the collar on the pipe was cracked. (My leak) So I have to replace P-trap and want to replace drain also.
Now I had several friends giving suggestions to remove nut on the bottom. It is very corroded. First thought was liquid wrench or something and channel locks. The nut looked to small to get a good grip. So I listened to suggestion number one, use a chisel and hammer, it will easily break off, not... Suggestion number two, cut it off with a dremel tool. Still trying, but blades keep breaking off. They are like ceramic or something. Not to mention I might of cut into the fiberglass on the bottom side of the tub. I hope it will still seal if I ever get this thing off. So, now twisting off is out of the question, because I have a mess of the threads. I'm thinking hack saw if I have enough room. So any suggestions for a mess??
I was going to add picture, but, do not see where I can.
Thanks,
Marianne

464954649646497

Well, we always take a small chisel and carefully tap the nut sideways, just like you would try to "unscrew" it. The chisel should be sharp so it cuts into the nut. The nut will eventually split in half.

Try again. Be patient. It will eventually go.

Milo