I'd be greatly surprised if this law had any noticeable effect on suicide rates.
I suspect it would prevent a fair number of suicides, although as noted above the price would be the inability for, say, a woman to quickly purchase a gun to defend herself from an abusive spouse.
As with most laws, there are both intended and unintended consequences.
I'm not doctrinaire when it comes to the Second Amendment. I don't think it's any different from other Constitutional rights in allowing for reasonable regulation. But such regulation, to be reasonable in my view, must be efficacious. And many proposed gun laws are of the feel-good variety; they appear to be "doing something" about gun violence but in reality have little impact except making it more difficult for the law-abiding to purchase a lawful product.
I evaluate each proposed gun law on its merits.