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).