Seen for sale that watch ( on chrono24 you will find it )
Caliber is certainly a P1008BS, I can't see the number bur surely 50920X, that is alright for a CR.
Case number 351155 ... well ok... but if the cylindrical part doesn't shock me, the lugs are quite surprisingm reminding some later references...
Now hands and dial are not what is normally associated with "Chronometre Royal" , I would suppose this is a franken coming with different parts especially for ahnds and dial... but that case? is there any reference with that sort of case?
(finally thanks for your help Alex ! )