I doubt it is coming out of the cable companies pocket, but through their insurance carrier. If it was a new car that was totaled, they would take possession of the total car and give you a check for full replacement value. Thirty-two states require car insurance companies to pay for the sales tax after you replace your crashed vehicle with a new or used one, but even then they often conveniently leave it out until they are sued for it.
Then they take the totaled car to a salvage auction and recoup what the can. They also can offer you to keep the damaged car with a lesser payout, to do what you wish.
I would think the insurance carrier would let you keep the TV and still pay full replacement value, but will need to be sued for the tax. Hopefully, they will see it your way and pay the tax as well.