Defeat to their argument.
This is not a straw argument, I am simply asking you to consider a stance if the oppositions arguments ended up being proven. It's not some complicated thing that you have to dodge the question by saying you would need to see the data first. It's basically one of three possibilities, banning all guns would either produce a decrease in deaths, an increase in deaths, or no significant change in deaths. It isn't a gotcha type question, it's meant to merely make you think through
When I look at how people argue the point of whether legal guns save more lives or cost more lives I often see pro gun rights people who almost cling to the idea that were it not for guns we would be awash in blood. They are very invested in the idea that guns can't possibly be playing a role in homicide and suicide rates to the point that they fail to be objective about the data and are letting their interest in firearms lead their conclusions instead of the data. For example I notice this when they all point towards the liberal estimate of 2 million crimes stopped per year instead of a more conservative and documented number of 60,000 instances per year according to the FBI (which by the way is still doubt the amount of gun deaths). Its not that I want guns to be banned, I want to be able to keep owning firearms, but I can see past that bias where sometimes the data supports guns and sometimes it doesn't. Some data does show/correlate with a reduction in homicide or suicide when guns are more strictly controlled and some other data swings the other way as well. The real scientific position is to look at both sides of the data.
However I am pointing out to you why it is not a key argument that we have to hold so dear to. We don't have to prove that guns don't cause more deaths because our interest, our justifications, and our rights to have guns do not rest on whether the guns can be proven to have a net lives saved or lives lost. Even if we could prove that a revocation of the 1st amendment would save more lives it is something I (and I assume you as well) would never give up. Since my assertion that guns are an important right does not hinge on whether there is a net life savings or a net lives lost I do not become so attached to this idea that I must defend at all costs the idea that guns being legal cannot have anything to do with our homicide and suicide rate problems.
You don't have to "think through" anything if you simply look at the reality that's in front of your eyes.
New York is reporting a 31% drop in GUN CRIMES even after a massive increase in LEGAL GUNS inline with the national numbers of new gun sales.
To state it plainly: MORE LEGALLY OWNED GUNS = FEWER GUN CRIMES.
Officials in NY are explaining the drop by pointing to the arrest and sentencing of ILLEGAL gun owners. That flies in the face of the anti-gunners' narrative completely. It also destroys your academic "thought experiment" here. Having more guns in the hands of law abiding citizens does NOT increase gun crimes.
There are too many examples of locations with increased gun ownership and lower gun crime stats to dismiss the probable causal effect more legal guns in the hands of citizens does not create more gun crimes. The opposite has been proven over and over.
I think what you and other "purely hypothetical" gun crime thinkers have in common is you only focus on one aspect of guns at a time. Instead, reality shows that while more guns sold might show a small increase in suicides (a victimless "crime") or domestic violence cases, the number is offset by increased defensive uses of guns as well as a reduction in criminals willing to take the risk of getting shot -- thus they find ways to get paid other than violent crimes.
I'd rather go with what's been demonstrated rather than trust the disproven theories of "thinkers."