Or in such a national case CYA to make sure he gets to the next state alive!
Ehh, not necessarily. If they don't give you a reason to, you generally don't as 'suicide watch' is generally a pretty miserable place to be and involves not giving that particular inmate the same rights as others. Generally that is an area that you tread lightly and don't just willy nilly put people on.
That garment he's wearing, that's ALL you get. You get no bedding, no other covering except maybe a blanket that looks just like that garment. no shoes, no underwear (yes, completely naked under that green thing) and usually no bed to lay on except maybe if the floor in the designated suicide area is 'padded'. No books, no papers, and your meals are fed with either a paper spoon, or a piece of celery.
All-in-all, the inmate has to give you a reason to, be it by statements or actions, and killing folks, isn't one of those actions, in general. Depends on how the jail's policy is written, but lots and lots of jails follow pretty standard policies, with small modifications made. You don't start making exceptions because you don't want to all of a sudden have to defend a non-standard procedure if something goes sideways.
Now, a statement can be as simple as something along the lines of 'y'all should just go ahead and kill me'. 99.9% of the time that one will get you there, usually they spend a few days on watch, and a mental heath practitioner comes along and clears them to population.