As much as I obviously detest guns and gun ownership, adding more restrictions wouldn't do a great deal to help reduce suicide rates.
I don't know, research suggests that removing a quick and easy means to commit suicide does reduce the suicide rate. Fot example, people have scoffed suicide nets on bridges because people would just find another method, but that does not appear to be the case. Also, when England changed its gas system so that it wasn't super easy, fast and painless to kill yourself by sticking your head in an oven, the suicide rate dropped noticeably rather than shifting to more difficult methods. Freakonomics (I think it was the podcast) did a really interesting dive into these numbers.