...easy to get at the movements on these...well, usually. I recently cleaned up a Pt JLC of similar construction. If you have the correct sized screwdriver, you can gently undo the screw in the middle of the crown. Once that is out, the crown comes off and you can (carefully) prise the caseback open. The hard part then will be to actually find the movement number on such a tiny thing. But it will also allow you the opportunity to clean up the case and dial.
I am on the search for a similar integrated bracelet watch for my better-half...I hope she doesn't see your post...or the pressure will be on.