I might be completely missing some basic understanding of your legal system, but isn't the Consitution a living document, at least to some degree?
Yes. When the government is working in the way intended, the Constitution is supposed to be a living document that changes over time, and that's why the ability to add Amendments exists.
The problem is, it has been a long time since Republicans and Democrats worked together to govern in the best interests of the American people. Since roughly the Reagan-era, it has been polarization and entrenchment, and any discussion, collaboration, or efforts to work with someone on the other side of the aisle has been seen as traitorous, rather than an effort to reach accord on real issues.
The last time we had an actual Amendment to the Constitution that actual changed any material part of the founding document, was in 1961 when we limited Presidents to two terms.
Prior to that it was giving women the right to vote in 1920.
So in theory the Constitution is a living document that should undergo changes from time to time to adapt to the changing world. In practice? Our political discourse is broken, and until that is repaired, changing the Constitution just isn't feasible.
...I think that a restriction on ammo and a reduction of supply might help quite a bit.
True. But people don't buy ammo to go on a gun rampage. They use what is available to them at the moment. 220m guns take a lot of ammo. And people have been stockpiling that shit for decades.
America is also fiercely anti-regulation, and it is an easy task to equate limiting ammo sales and production with "killing American jobs". Ain't nobody gonna vote for something that puts Americans out of work.
Again, like I said, this is a social problem to be solved with treatment programs. Making suicide a guns issue is failing those who are inclined to commit suicide. We need ways for them to easily get the help they need, when they need it, whether they want it or not.
Small anecdote: I lost 5 friends to a mentally-ill man with a gun, who walked into their cafe and shot everyone inside. His family knew he was unwell. They had tried a variety of treatments. But he was an adult, and living in a different state, and the laws said you had to consent to being committed for a psych eval, and he - being sick - refused the service. Multiple times.
His family knew he was sick. They appealed to the authorities to help them do something about their son/brother, and the authorities failed.
Blaming the gun for his death is a disservice to him and to anyone suffering from depression, or any mental illness. We need to address this problem systematically, with better options available all along the line.
Unfortunately, Reagan dismantled these programs in the late 1980's, and there is no way to rebuild a social-service program in America, now that everything has to turn a profit.