I don't think you could land a job, per se, but you could possibly create a practice. I don't know the demand, that would be learned as part of your business plan's market analysis. I doubt it would have anything to do with passing Japanese tests since you wouldn't be working with Japanese people except under extraordinary circumstances (read: practically never). It also wouldn't be based on a 4-year degree, you would need the same qualifications as you would in the USA if you wanted to do clinical work (PsyD or PhD, I think) because your clients would want to know that they're seeing someone qualified. You would need to do the research to see what you can and cannot do in regards to the law, Mike's link has good info on that but if it were me I'd pay someone whose job is to advise on these things before I put the kind of required time and money into a plan like this.
For me, in a somewhat related health field, I cannot take NHI since I do not have Japanese licensure. I work almost solely with expats, and I work solely in English because it would be unethical to work in Japanese without significantly better Japanese than I have (frankly, significantly better than N1). I had legal counsel vet my plan to ensure that everything I am doing is within the law. It cost a fortune monetarily and in time and effort.
Frankly, this is not an easy way to get to Japan.
A 4-year degree in psych is an incredibly poor decision in the USA, and probably would only qualify you to get a job teaching English here.