I've only been working on carbines for about 10 years so I am not your best authority, but if you refinished the stock it may be a very nice and relatively valuable stock, but it is no longer collector grade. Therefore, if you intend to shoot the carbine and the NEW recoil plate screw will not draw down sufficiently tight to protect the stock from damage then I would very carefully attempt to replace the escutcheon as described above. Worst case scenaro, if you do chip out the excutcheon hole you can repair it by glueing back the original wood chip and refinish so it is barely noticiable.
I definitely would do nothing until I tried a brand new recoil plate screw. For the life of me I don't understand why either the screw or the escutcheon threads ever wear/strip but I do know this appears occassionally. More knowledeable members may be able to offer better advice on this matter, both technical and non-technical, but I think we will all agree that if you must replace the escutcheon nut you should do it very carefully to protect and conserve the stock as much as possible. Good luck.