Wiring for a ceiling fan. Do I create a switch or can I hard wire directly from a wall outlet?
Can’t power bathroom receps from non-bathroom circuits
A recep in a bathroom is only allowed to draw power from 2 kinds of circuits, and this is a very narrow definition:

A.  Circuits which only power bathroom receptacles (in any number of bathrooms)
B.  Circuits which only power recep or hardwired loads in this bathroom

Any circuit that powers anything in a non-bathroom is instantly disqualified.
Your bathroom may already violate this rule, and as such, may be grandfathered.  The rule with grandfathering is you’re not allowed to make the situation worse.  So you can draw off an existing bathroom recep, even if it already violates the rule... but you cannot add a new violation by drawing off a bedroom recep.
As long as you have a controlling means for the fan that you consider reasonable (are you OK with your partner calling an electrician to unwire it if the remote breaks?) then you are all set.  I for one would feel better having some sort of disconnecting means.