That makes them extra valuable. I hope you remind your insurance company of this.
We sort of did this with one of our cars. The insurance company wanted to declare it totaled. Instead, we asked for and got the amount they were going to give us minus the wreck value of the car. We had to sign a waiver not to seek any more money. We got the car fixed for that amount. It was just some smushed-in front damage.
That was something like 9 years ago and I’m still driving it.
Note: You do this before having the car officially declared totaled. Once the car title says “totaled”, it’s a whole 'nother thing.
It shouldn’t that quick, but if you have doubts, just back it into your garage/driveway/parking spot for the next day or two. That way the hood* is always out & reachable via normal-length jumper cables to another vehicle.
*Assuming this is a normal car with the battery in the front.
So is this something I would tell the adjuster when he evaluates the damage? Does he make that call or just send a report of his estimate to the insurance company? The car is in excellent condition and has never had any repairs. When I bought it 3 years ago it only had 19,000 miles on it and now only has about 30,000. I was planning to drive this car for years. It would be extremely difficult to find another deal like that.
There is no way I could back up my driveway. It goes up a hill and then makes a turn to the right. Driving up it, you can’t see the turn and after 15 years I still misjudge it. There’s a permanent rut in the yard on the left side of the driveway from years of this. None of my friends will even attempt it to come up it. 
It’s the most ridiculous driveway ever made. So I’m parking on the street for now.
In the states, the laws are different from state to state. In my current state of Georgia, a totaled car gets a “Salvage Title.” If you decide to keep it, you have to take before repaired pictures, keep receipts for any and all parts or work done(by a licensed re-builder,) and have the repaired vehicle inspected. If it passes, you’ll get a “Rebuilt Title,” allowing it to be registered again. Check your local DMV/Tag Office/whatever it’s called in your area.
Just a quick tip about batteries in general. If you’re worried about your battery possibly being drained, you can pick-up a cheap volt meter for around ten bucks. Just take readings for about a week. A fully charged 12V battery should read around 12.6V and shouldn’t drop more then about .01-.02V per day at most. I suggest this instead of unhooking the battery right away because a modern car’s computer hold lots of data that can be beneficial for any future repairs. Unhook the battery for too long, there goes all that info.
What is with certain states and these inspections? If it passes emissions testing, has wheels and lights that work, what is the problem? Yeah, sure safety and all of that fun, but why is it always a “licensed” repair is necessary? If you fix it yourself and it passes an inspection, that should be enough. I suspect lobbyists.
Unfortunately, it’s probably the other way around. Unless it’s a sports car or something like that, a manual transmission makes it less valuable (however hard they are to find, they’re evidently even harder to sell).
When my truck was struck 3 years ago, I was given two offers by the other driver’s insurance company, a “totaled” vehicle price based on the book value, or a lesser, -you can keep the truck-, offer. Difference was roughly twice the scrap value of the 10 year old truck. There was drive train and frame damage. And I’d still have had a rusty truck with holes in the box.