Wow this is cool! I have a similar tub, circa 1920, kohler, it's in great shape after 100 years. Looks like you updated the tile and some of the hardware. I have not yet tried to remove the drain grid, mine appears to be held in place by a center screw. First question, do you have access from below? The other is, have you tried using a tub drain wrench to apply torque? The brass is somewhat softer, so it might help spread out the force, but you'll have to try to be careful not to damage it. Obviously the correct fix is to remove the drain and re-set it with plumbers putty or, I believe Terry used to recommend silicone on tub drains where there is no access to change them anyway, but with an antique tub in this case, I think I'd go with the putty to make it easier to service. Perhaps there is ad adhesive sealant in place from the last repair, maybe you could try to cut it with fishing line or dental floss? If you do manage to get it out, and you like your tub, it might be worth have it re-chromed while you have it out.
As for caulking around the edge, it might work, but it obviously isn't the ideal repair. However if this were done and it worked, I suspect it would hold up at most 5-10 years and maybe less. I would try to use a specialty caulk that will stick to it's own old caulk residue, unlike ordinary silicone, because it will be easier to repair or replace given the shape and limited access you have to clean the joint.
