Bathroom GFCI trips when I plug into different outlet on separate breaker, but in same bathroom
The neutral on the other circuit is somehow cross-connected to this neutral.  Possibly via bootlegging grounds. Possibly via a miswired light or fan.  

All this miswiring is related to the LOAD terminals on the GFCI. Those are shipped with a piece of tape on them that says "Do Not Use.  For Wizards Only."  But lots of people don't read the tape.  Especially selling homeowners trying to pass inspection with GFCIs in all the right places.  

The usual mistake is replacing a plain receptacle (which has 4 screws, meant for splicing) with a GFCI (2 screws, plus 2 special Load screws not for splicing). They just force it, and it causes things like this.

So the answer is: 


search all boxes in the bathroom and remove any bootlegged grounds. If they are GFCI protected they are allowed to be 3-prong sockets and have no ground, but they need a sticker that says "No Equipment Ground".
remove all the wires from the GFCI Load terminals, and place them on the GFCI Line terminals.  Use any splicing method that is legal, e.g. wire nuts and pigtails. 
understand your wiring.  If you know of any particular loads that you want GFCI protected, move their hot and neutral to the Load terminals.  Do this symmetrically, don't move hot while you get neutral from somewhere else.