The screws that hold the plate that covers the drain toggle switch thing in my kids' bathtub rusted out. One of the heads of the bolts fell off and when when I removed the cover I resized the other one wasn't actually attached to the bolt either. I squirrels a good amount of PB blaster in there and tried to use a bolt extractor but can't make a dent in the bolts. It looks like they're fused with the cast iron drain.

Any ideas?

Try ice on the bolt head and a butane torch on the interior of the drain hole and pliers to try and budge it?

Have access behind tub to replace entire drain? Even if it takes a drywall cut, that's likely going to save yourself so much hassle and time even including the patching and painting after. Also, tbh, it looks like the old drain has been allowing water to drain into the wall around the outside. Probably want to fix that either way.

That drain tube is probably brass, which means it's soft. To get those screw out, if you can get them out, here is what I would do and use.

I'd try to grind or file the screws flush with the drain tube. I'd do this to give me the best chance of centering my set punch. Then, I'd use my set punch to put a divot in the center of the rusted screws. That will allow me to have a clean and easy spot to rest my bit to drill a pilot hole. I would use a set of COBALT bits for this. They are hard and I've extracted stainless bolts with them. Brand new bits. Then try to get as square as you can. If you cannot get an ease it on it, then you aren't going to have a lot of options left short of replacing the drain tube in which case I'd toss everything and get a new kit for that.

There aren't many techniques or magic beans for removing seized screws that have nothing to bite into. It's drill and extract and that's about it.

If you can't extract then try to get the proper bit so you can drill a hole and tap a new thread. That part is an overflow cover and it's not like it's a structural component or holding the ailerons on the space shuttle. Even a half baked drill and tap will hold.