Pretty sure the US Constitution trumps the management company handbook and lease document.
Actually it doesn't. The Constitution prohibits the government from abridging your right to keep and bear arms, but does not prohibit a property owner from infringing on said rights on their tenants. Now if it wasn't in the existing lease, they can't change the terms of the lease until it comes up for renewal. But on all new leases they can impose such a restrictions.
Frankly I would ignore such an idiotic provision if I lived there. Their only recourse if they find out you have guns is to evict you. And how are they going to find the guns if you don't tell them you have them?
This is just a publicity stunt and it is going to give them a lot of publicity. Bad publicity. They will lose a lot of good tenants over this and they will exchange them for really really really bad tenants.
It’s a private company, not the government.