A guy I worked with had his worthless 80's Volkswagen totaled by someone else. He had rebuilt everything in it. He said since it wasn't his fault, he was going to get all the value out of that car and enough money to rebuild another one. He of course had to rebuff the $800 offer to total it. He told the insurance all his work, documented what he could, and told them he expected to get back what he had in it. And they reimbursed him significantly.
I've done the same thing a few times, once for myself, and a couple times helping out friends. If you can talk to an adjuster in person and actually point things out and show them - so they know you're not BSing, they've been very helpful in my 3 cases. You do have to get through the initial work of convincing them you actually did a lot of work to this car, above and beyond just repairs and inspection costs and break downs - I'm sure they hear that all the time, so don't just expect them to believe or trust that what you say is above and beyond everyone else they talk to. Once they realize you're not average Joe, they've been really helpful to me. and I think avoiding a lawyer (which I've always done) is helpful too. But none of those incidents involved medical issues, so that simplified things.
I'm sure I got lucky with the company and person I talked with to some extent. But try to get face to face and point out and show you're not like everyone else just trying for a money grab and maybe you'll get lucky. I've always had it in mind that if they tried to not play nice, I may have to drop hints that I might have to escalate or get some help settling this (lawyer hints), but I've never needed to do that. They let me keep my totaled XT6, without totaling it so i didn't have to juggle the title/DMV reinspection, and also cut me a sizable check for far more than the "total" value, when a lady hit me. They're generally limited to totaling a car, or not, but they have ways of going above and beyond that if it's in their best interests which I would guess it is if we're avoiding a lawyer.