Looking at the BYU website, I think I would talk with students who run the Fencing Club, too. BYU doesn't have a lot of club teams listed, but it looks like the fencers compete regularly. They can give you the inside scoop on working with whatever office oversees student organizations. It looks like their email addresses are linked to their webpage. You will need to figure out the pool time problem.
Some clubs are required to pay for pool time, others can just request it.
You will want to talk with the pool manager about finding a slow time when you can get deep water and enough space to do swim-throughs regularly. You may have to start with a corner of the pool during an open swim time and build up from there. The PE department offers swimming classes, including WSI and lifeguarding. Talk with the aquatics instructors about synchro. You may find someone on the PE faculty who did synchro in the 1960s or 1970s who can support your efforts to introduce the sport.