Seems like you need a good vintage car insurance company! You got to do some thorough research before you make a decision. Having to keep the car in a locked garage is a huge expense!
I think you should definitely up your insurance coverage; 12K doesn’t begin to cover what that thing is worth. Prepare for an increase in auto insurance rates if you do it; personally I think a little extra is totally worth it though.
See if The Hartford will give you decent classic car insurance quotes; I know that they dropped my dad’s coverage down to comprehensive when he garaged it. I believe they’ll tailor your policy to your needs so you might be able to get what you need.