I have several bookmarks left in the yard trees I regularly climb. The ones I use most often are #18 (200lb) braided nylon net twine from Memphis Twine. Some are over two years old, and still work fine for pulling my climbing rope seventy-five feet up. It's pricey, but not as much as good throw line.
The other bookmarks are twisted nylon mason's twine I get out of the trash at the local football stadium after it's been used for marking the field. It also seems to last a long time, but it's not as strong as the braided nylon. If the tree has one of my permanent friction savers, I'll use it to pull the rope up. If there's no friction saver, I use it to pull up a braided nylon line or an old throw line first.
I just leave the bookmark line on the ground while I'm climbing since I'll be putting it back up when I'm done. Be careful when the rope end approaches the friction saver. The rope will start falling of its own weight, and the small line will get tangled in branches or pulled out of reach.
There are oodles of squirrels here, but they've never bothered my bookmarks. I don't use gloves to handle the line despite my hands being sweaty. I've had squirrels eat pvc conduit and automotive wiring, but never nylon line.
I'd stay away from the stainless cable, especially the very flexible 7x19 type. It's more likely to have broken strands, and they're hard to see and incredibly sharp. I always wear heavy leather gloves when handling cable. A cable with a broken strand could certainly do serious damage to a climbing line.