I don't know, it seems to work OK on my RX100 mk I
I just select M on the top dial, then press the left side of the rear control wheel to access drive modes. There's no apparent restriction, it even seems to work in Bulb mode.
I doubt that the menu system differs between the RX100 variants in this respect, so something else may be restricting drive mode selection. I can't see what though, as the manual suggests that the drive modes available in PASM modes are the same. The only function you lose with M mode is Exposure Compensation.
Welcome to the site, btw.
-R