Hi, I have a bachelors in psych and a masters degree in human relations ("professional psychology concentration"), both degrees from a major university, am a licensed marriage and family therapist (LMFT) and have been since 1991. I minored in social work as an undergrad. I just graduated from nursing school (last Friday) and am getting ready to take my boards.
I've been researching this idea of psychiatric nursing with an eye to private practice, which sounds a lot like what you are considering.
If I had it to do over again, and if I were at the cusp of my first degree, as you are, and if I wanted to do what you describe, I would switch my major to social work and/or get an MSW. It's about the same amount of time as the RN would be, according to the plan you outlined (RN after you get your BS Psych).
The MSW program is not terribly challenging, I think your worst problem would be boredom. You will thrive in the counseling atmosphere, you will have a credential that opens a lot of doors, gets you reimbursement from everybody (this makes a lot of sense in private practice--glad to "talk" about it more if you are not familiar with it), and you don't have to do the prereq sciences, and risk not being accepted into crowded programs.
Don't get me wrong, I love nursing, and it's really where I belonged from the get go. But rather than get my master's even in psychology, I'd've gone the social work route.
The only thing you would not be able to do as an MSW that you could do as a psychiatric NP would be prescribe, and in some states you couldn't really do that.
Good question, hope this helps. Feel free to email me if you like.