It may be worth your time and effort to plead your case. You might find helpful info on the DOT website in your state on the matter.

You might also try contacting your state representative, or a more local political entity. I had good luck with my township commissioner and state rep who got the PA Dept. of Transportation to fix an issue they originally failed to do when I had tried contacting them directly.

I doubt the state will buy you a 15' tree. You could probably get them to remove the tree, though, since they killed it. Get the local arborist you spoke to to send them a letter with their opinion about the tree. (That the cutting caused the tree's demise and that the tree is dying/will not recover and needs to be taken down.) Include lots of photos. If you give them a quote for removal and a new tree, you might be able to get them to reimburse you if you can get a good deal on the tree, but I wouldn't hold my breath.

You might want to check who actually owns the tree if it was close enough to the road to be cut. I know around here the trees right on the road belong to the town. If it's their (town/city/state) tree, it's their responsibility to take it down. Also, you might point out if the tree falling or dropping branches will impact the road.

Our town actually wouldn't let us take down a mostly dead tree because it was a "town shade tree" (it was hollow and had one semi-leafed branch after being hit by lightning twice ... not sure where the shade was ...). We finally talked to the tree warden enough that it was removed. They did consider planting a new town shade tree, but decided not to because of a nearby giant oak and the location of the gas line.

Oh, and on the cut out the dead vs cut it down ... state isn't going to pay twice ... so take that into account, but in terms of just replacing the tree:

When we had a dying apple tree in our back yard, we had the dead parts removed and planted a young serviceberry nearby, so the serviceberry could start establishing and growing before the apple was totally gone.

A 15' tree isn't going to shade that whole side of your house anytime soon either. Yet cost significantly more than one in a normal garden center size. At the same time that big transplants may often also be less quick to bounce back after being moved because of a tendency to dig them with a lower proportion of roots retained than with smaller stock.

We lost a tree to the highway department. While we considered the tree to be part of the landscape of our lot the highway department had other opinions. The house was about 80 feet from the house on the side road on that side of the lot. We were working to get that part of the yard a forested park like area to complement our house. The tree that was lost was part of the long term plan.

While we did get something for the tree, it was much less that we had hoped for, as while we considered it a yard tree the highway department considered a forest tree.

I thought about planting a smaller tree, but it appears I’ve run out of time. :(

Our house stays in the 75-78 range even in July because of the shade from the maple (we live in central NY). We have hydronic heat, and the main feed into our house is only 100amp so installing central a/c isnt an easy option.

This picture is from right after the tree was cut. There were dead branches before they cut the tree; the states trimming accelerated the process.

Consider a "whole house fan", one where the motor is located remotely. Done properly, you'll need extra attic vents etc.. It won't lower the humidity but you will feel cooler.

https://www.energy.gov/energysaver/heat-and-cool/home-cooling-systems/cooling-whole-house-fan

They are tough trees! (large Maples, generally) One of my neighbor's Norway maples, I think, had the ground under it regraded by a careless contractor. I told him they had piled up too much dirt over its roots and he didn't believe me. Looked like 6-8 inches over 75% of its root zone.
Well for 2-3 years the canopy was thinning and I thought it was going to die. But it seems to have recovered.

Are you sure it was the state that trimmed the tree, or was it possibly the power company clearing the overhead wires?

Did they trim the top or the roots? I would call the County road dept. to find out if they have an arborist who could come look at the tree and give you a recommendation on how to save the tree. It would be a shame to lose such a beautiful tree.

Tree that size is going to take decades to be matched by a replacement. Regardless of how big the replacement is at planting. Unless you spend say, $20,000 on having an unusually large transplant brought in using a tractor trailer and so on.