Are you saying it would be better for me to take that now rather than be forced to pay an unknown price at lease return since I would have no choice at that point?
I also plan on purchasing a Tesla after this lease. Not sure if this would put anything in my favor about them looking the other way on this (probably not).
It doesn't work that way. A couple of months before the lease ends, you'll take a bunch of pictures and send them to the lease returns dept. They'll examine the pictures and send you an estimate. If you still have the damaged seat covers, you'll have to send them pictures of the damage so they can come up with an estimate. As long as you are good about taking pictures, that's what they'll charge you at lease end.
For some items, having tesla do the repair may be cheaper. For instance, they charge $85 per wheel for wheel rash repair. I couldn't find a shop that would fix them for less than $125 each, so it would have made sense to have tesla fix them and pay the $85+. However, I bought a $40 bottle of wheel touch up paint and completely repaired one of my wheels that had a little rash and touched up another that had pretty bad curb rash. Ultimately tesla determined that my bad wheel was good enough and refunded my the $85 + tax I had paid. This is a BIG YMMV!
For tires, they only put the oem tires back on. They quoted 4 tires for a total of $1340, (+ tax I'd guess). I couldn't install 4 oem tires for that little, but I did find a deal on some closeout tires at tirerack that met tesla's specs for about $750 installed. That's what I did. They had no problem with them.
Tesla could care less if you are buying or leasing another tesla or walking away. They are going to charge you the same either way. However if you do lease another one, they will waive the disposition fee. (I'm not sure about buying). If your sales advisor tells you they will waive damage if you do, get that in writing. My guess is you won't see that email.
If you do need new tires, do them 6 months or so before the lease ends so you'll get to enjoy them a little. I wish I did that.
Tl;dr: take really good pictures and the estimate you get is what you'll pay.
Since you have already consigened yourself to replacing the seat cover, I'd get a little more aggressive first. Maybe some white shoe polish. I think the white wall tire bleach is a really good suggestion too.
Good luck.