I don’t know Richmond well, but I think that most schools allow self-designed majors, and a quick google indicates that they do: http://undergraduatecatalog.richmond.edu/curriculum/as-programs/department/interdisciplinarystudies.html
Bummer about the NPCs, but good that you determined this sooner rather than later. The $20-$25k price point is one I know very, very well 
but we were lucky that the NPCs agreed with us. Having been through this just last year - and my kid applied to Rochester, Northeastern, Tulane and Case - I think the answer to your original question is no, there aren’t any privates where you have a ‘decent chance’, interpreting that as meaning ‘reasonably likely’, to get a full-tuition scholarship. You’ll be in the running, but they’re extremely competitive, and often require separate applications and essays, candidate fly-in weekends and such. Northeastern University Scholars is one of the few that doesn’t. The full-tuition scholarships often go to rock-star kids who would have been competitive for the tippy-top schools but looked elsewhere because those offer only need-based aid for which their families weren’t eligible.
Are you male or female? Would you consider a liberal arts college?