I think you have a chance and only Rice may say no. May because you are far above your school.
But you need aid. Did you run the NPCs of all to ensure they’ll meet your budget ? This is the most important thing. And if they do and it’s a lot, you might get caught up in a need aware process (if you need too much). So you might consider adding need blind privates depending on your budget because let’s say you need $75k, a need aware school may reject you for that reason alone.
Congrats on a fine record and best of luck.