OK hear me out, I just got a new laptop and been using the shitty keyboard cover which it came it.
As it was kinda nasty I went looking for a new one, had my mouse mat on my left and though "uh if only u were the right size and a bit slimmer" and an idea just ringed my hears
"Laptop keyboard cover that could double duty as a mouse pad"
Soft surface and grippy edges so it wouldn't slide, washable so u can get ridd of the nastyness of the tables u bring it to.
Currently I carry everyday my laptop, my mouse and the charger. If bringing a mousepad was as simple as removing the keyboard cover and putting it in my desk I would pay a couple bucks for it.
Keyboards covers are usually too thin to be a decent mousepad.
Id look for a mousepad that fits into your current bag so probably something the size of the laptop itself
I'm assuming that OP is referring to the white (sometimes foam, sometimes shitty microfiber) packing material that is placed between the keyboard and screen for shipping. Since most mice are optical based the tracking surface doesn't really matter unless you are someone who needs extreme precision so this "keyboard cover" would probably be fine.
Sounds like a terrible idea - mouse pads get dirty, on both sides, and you'll be transferring that dirt to the screen.
There's really no safe way to clean a laptop screen - they generally have anti-reflective coatings which will be worn down when you clean them, and it won't wear evenly it will be worse where you've cleaned off a particularly bad smudge. I try to be as careful as I can but still all my old laptop screens are pretty bad by the time I upgrade and I feel like your proposed idea would make that even worse.
What you can buy is a laptop bag/sleeve that doubles as a mouse pad. Some of them can also lift the screen up high for better ergonomics - though then you'd also need a portable keyboard as you definitely don't want that up high if you're worried about ergonomics (I recommend the Nuphy Air if you're interested in a good portable keyboard).