I think he may have a crush on her by the sounds of it, at the very least.
I would be fine with him not wanting her left on shift alone at night, if it was then being dealt with as a staffing issue in work, as in they need to pay someone to do that if another member of staff is required.
I would be fine with him texting her as friends/work colleagues. I had a male work collegue who was then a friend.
Even buying a present for the child and perfume for her wouldn't really bother me.
I would think it strange that he wants her to carry around something from him, that is strange IMO
When you put all the above togethet though,i think its a bit of infatuation
Im so sorry you are dealing with this just 5 weeks after a baby.
LTB, on its own, is not helpful advice with 2 children and a 5 week old baby.
You really need to get to the crux of what is happening, he needs to man up and be honest.
If it were me, i would rock up on his night shift and have a very open discussion, with the both of them.
You may well find out she absolutely does not appreciate these advances and hes being a sleeze or he very well may be telling truth, but its a shame it would come to that but you need to know the truth and if you dont believe what hes saying, go to the source.