No insurance company will touch this. For a car, the damage is quantifiable as in cost of damages, replacement etc and value of premiums are tailored to factor in age, vehicle type, driver quality etc. None of those are available for guns. Also imagine families of victims lodging a claim on insurance company for value of lives lost....how does one even put a monetary value to that?
You can consider something like a Gun tax or something. If already exists, then may be hike it drastically. I just looked it up and an AR-15 seems to sell for about $500 average. Have a 25% of value tax for all small guns (hand guns etc) and increase it to 50% if it's a higher class of gun (semis, AR15s etc).
You underestimate the ingeniuity of acturial mathematicians/scientists and/or quants at quantifying risk. Every obscure project/product gets insured, from offshore oil rigs, to CDOs, to celebrity property and movement, to festivals and raves, etc... It's a complicated problem, but to some measure the actual cost to society can be captured in a figure, and flowed down to the risk pool of gun-owners.
The gun tax doesn't solve the issue. It's a one time cost hit. More than paying for the actual cost of having guns in society, forcing the gun-owners to bear the cost of guns dis-incentivizes gun hoarding. If you have 30 guns, now you have to shell out $12000 per month for coverage. Shit, now you can't afford to hold them. You sell them off, or otherwise dispose of them. People who would otherwise buy a gun now have to factor in annual costs, and decide they can't afford to have one. As the pool of gun-owners decrease in size, premiums go up as insurers adjust. Eventually the equibrilium point is a small number of rich folk like Dan Bilzerian who can afford to pay the premium for a small arsenal, and the vast majority of Americans that have their small revolvers in the cupboard, and pay for it.
Impose punitive fines on anyone found with an un-insured weapon. If they can't pay, seize the weapon and destroy it. If you decide to run the risk of not insuring your gun, you'll hide it where it can't be easily accessed or found. Require any gun purchase to come with proof of insurance (effectively creating a record trail and closing the private sale loophole). Insurance lapses get reported to the ATF or state equivalent, the same way Geico reports car insurance lapses to the department of motor vehicles. Criminals can't go to Kentucky now and buy loads of guns and drive them up to Chicago or NY, as proof of insurance is required at POS.
Finally, make every gun owner feel the pain of every mass shooting through a rise in rates. If Geico determines that men below the age of 30 are causing more accidents, regardless of how well I drive, my rates are going up. Hurt the pocket of every responsible gun owner. Eventually some will decide it's just not worth it to have a gun in the house.
The legal and infrastructural challenges would be huge. But creating financial incentive for not owning weapons would drive feedback loops that would eventually stem the tide of the gun problem.
I'm sure I'm missing something and I just wasted 15 minutes typing this.