There aren’t typically intentional efforts to weed out at any school. Schools where you see higher levels of attrition are state schools that accept students that aren’t ready for the rigors of engineering.
Don’t worry about lack of exposure to FIRST, etc. Most students that enter ME programs don’t have any previos experience, but rather an affinity to math and physics. In fact, students that love FIRST, but struggle in math and physics are actually the ones more likely to drop out of engineering. Engineering can have facets of tinkering, especially in concentrations like mechatronics, but at its core it is applied math. Our son is just about to finish his MS in ME and he never had any prior engineering exposure before starting in college.
As far as merit goes, you are going to be limited. The purpose of merit is not to reward students, but to entice students with high stats to attend their school to raise their rankings. Thus, either his SAT needs to come up, substantially, or (and there’s nothing wrong with this) he needs to consider non-selective schools.
A school that I would highly recommend, but probably would fall outside of the financial budget is WPI. Depending on your eligibility for need based aid, it might come in line. Like all private schools though it’s about $70k per year all in before aid.
I’ve also read great things about Miami of Ohio. I wouldn’t write off Binghamton or Buffalo either. I second SD and NM, but they are both in the boons. It would take a special personality to leave the city and feel that either of those were a fit. In the same realm, Wyoming is an underrated engineering program and VERY affordable.