Bummer!!
Unfortunately if mouse milk doesn't do it nothing penetrating
oil wise Nothing will.
Looks like heat is the go for a start.
If that doesn't work drilling will be really difficult with the stainless because it work hardens real easy.
Take your time with a new good quality drill,
oil and slow speed.
If you have good access try drilling dead centre if you can with a drill the I.D of the thread then sometimes you can then wind the thread piece left over out with pliers.
Good luck