I agree with your designer that they all should be the same. However, I am sure some people will post that they have a mixture of painted and stained trim. Some people have done this but the norm is to go all painted or all stained.

White trim in the bathroom is fine. Again, using my 80 year old previous residence as an example - the two front rooms, living and dining, had oak floors, doors, a built-in buffet, windows and trim including a 10" baseboard. The rear 1st floor rooms had maple floors, cabinets, doors, windows and trim. Upstairs the floors and trim were pine. Outside the wood was all painted the same. Nowadays most people do use all one or the other, stained or painted. But you can do it the way you want. My bathroom door was stained on the outside and painted white on the inside. It worked very well for the space and that house.