In a lease you are not paying towards any equity, you are basically paying to rent it for a term. The car is actually losing value as you drive it around during your lease. It’s less complicated to think of it in those terms.
Since you can no longer turn in the leased car at the end of the lease, you have no value in that car. There is no car. You cannot negotiate anything.
You would have had an OPTION to buy the car after your lease expires.
When you turn in a lease the dealership will go over that car with a fine tooth comb and ding you for every teeny cosmetic flaw, and even remaining tire life expectancy. Even a small stain in the carpet inside of the trunk will cost you. Even if they can steam it out.
Your auto insurance should be covering the cost of the car. At three years old, it shouldn’t be very up-side down. As others have mentioned - gap insurance is usually offered when one purchases new, and/or when one leases a new vehicle. First of all, see if you took that option. Hopefully so.
Lease payments do not build equity in the way that a straight purchase loan does. Except for the dealership, they make out well with a lease turn-in. Leases are best for people who like to trade cars every two to three years, people that only need a car for a few years for what ever reason, company cars, and people that need a less expensive car that is new and (hopefully) very dependable - some parents do this for their teens.
Hopefully insurance takes care of the worst/all of it for you.
Where do you get the idea that you have even one cent of “equity” in the car? If it’s a straightforward lease, you own nothing. You haven’t been buying a percentage of the car. You’ve just been paying for the privilege of driving it.
To use an imperfect analogy, if you go “halves” with someone on a five-dollar lottery ticket that hits for a million bucks, you own half a million of it. If you lend your friend $2.50 so he can buy the lottery ticket (to make it simple let’s say he just asks you for the money and doesn’t even say what he’s going to use if for) and he wins a million, you get the $2.50 back.