Either of the degrees you mention would allow for the career path and interests you've expressed. The specifics will vary per state legislative guidelines, but school psychologists can typically work in private practice if they've obtained the appropriate internship and, if necessary, postdoctoral supervision. Obtaining doctoral-level BCBA training and certification (i.e., BCBA-D) might help even more in terms of practice-related options and desirability. And doctoral-level school psychologists commonly work in academia, but this is something you'd likely need to plan ahead for while in graduate school if you want to truly be competitive for tenured positions.
Clinical psychology programs, assuming they're APA-accredited, should all prepare you for state licensure as a psychologist, which would of course allow for private practice.
I would agree that clinical students will have more options for APA-accredited internships through APPIC, but there are still a fair number that are open to school psychology trainees. If you want private practice as an eventual option, this will be a must.
Given your existing degree and clinical experiences, it sounds like a school psychology Ph.D. program might be a better fit overall. If you don't have much in the way of research experience, either degree will be tough in terms of admission. However, your current credentials would probably be more beneficial for a school psych program. You may still need to spend time gaining related research experience, but perhaps less so than if you were to opt for clinical.
Also, I agree with the above expert that posting in the psychology forum, if you're comfortable with that, would be helpful.
Good luck!