Why do you have a babymoon in a hotel?
This term, which has been continually in use since the 1970s at least, refers to the bonding period you have with a newborn, and always used to mean cocooning at home until you got to know each other a bit better.
If you would really want to do this in a hotel (and I can see the attractions if room service!) you will need to think about
a) what baby equipment you will be taking wi you, and what you are happy for to take a chance on at destination
b) location of nearest medical services (MW visit in first 10 days, and HV thereafter, but would expect to come to your home, I don't know how you book out of area)
c) flexible booking, in case your baby arrives early or late
I strongly recommend you rule out the Carribean, or indeed anywhere that would normally require travellers to have had jabs your baby will be too young for. Though there is no bar to going anywhere as soon as the baby has a passport.
Wherever you end up, I hope it is indeed the bonding process for the three of you that it's always meant to be.