Would love one of these myself, but reckon I'd get too much hassle from SWMBO about it, she was annoyed enough at the "old man car" S-type!
With the key fob, have you tried to re-pair it to the car? Jag have some weird methods, I think in the S-type you had to turn the key from 0 to 2 in the ignition about 10 times whilst holding the brake, then press another button and it'd re-pair. In the X350s I have a funny feeling its something odd like holding a climate control button whilst flashing the headlights 10 times, or some other bizzare combo🤣 but it might do the trick!