The rule of thumb for “teaching” (in quotes for the different types of teaching) at a college...
If it’s an official course for college credit, such as a PE elective, they’ll require a minimum of a master’s degree in a relevant field or comparable experience. Relevant field in this case would typically be physical education, kinesiology, exercise physiology, etc., or adult education. Comparable experience could be, say, a former pro dancer who’s successfully run their own dance school for several years.
Running an extracurricular club where credits aren’t earned allows far more leeway. Yes this is teaching, but not really in their eyes. They’d typically require references, background check, CPR/First Aid (for physical activities), stuff like that. Private schools have more leeway, and state schools have more red tape typically. With RPI’s reputation, I’d anticipate at least as many hoops to jump through, if not more.
To give you a sense of RPI’s ways, I applied for an athletic training (sports medicine) position 5 years ago. My interview was about 5 hours total after all was said and done. Individually, I met with HR, the athletic director, the staff I’d be working with, the head athletic trainer, 2 head coaches, a committee of employees outside of athletics, and a group of student athletes. At least they fed me lunch in the middle. All that for a position where I wasn’t a supervisor. And I didn’t get the job. I’m not bitter though.
If the OP obtains any credential certificate, it WILL be verified. If it’s issued by a teacher, they’ll look up the school to make sure it’s actually a school, and they’ll contact the person in charge. If not prior to hiring, then soon thereafter. Safety of the students is their primary concern.
I’ve seen a few people seemingly slip through the cracks only to be escorted off campus by security shortly after.
Please note I’m not saying the OP is someone who’d falsify any credentials, has a questionable past, etc. I’m just saying how colleges operate. I’ve had several friends in college HR. Unfortunately none at the college in question to be able to make a phone call.