Yes show them your offers. It’s up to them to decide if this is enough to give them more. Honestly for financials I think it’s best for the parents to do this. Schools don’t want to lose students if their close on the offers. Also ask for a housing grant /credit. Lots of schools have them but don’t advertise them.
One way to start is to say “I really want to attend but I need help financially to attend. I have some other offers. What can we do today to make X school a reality for me today”. Now the pressure and ball is on them not to lose you.
As stated they should be peer schools and better if they are in the same competing region for students. You also have more leverage if you are in demand. Like an athlete or a in demand theater person.