My current go-to is "I'd love to, but I can't because I'm folding t-shirts at that time". Humorous rejection for those who get it, passes the photocopier 'because' test which actually suffices for many, and achieves your confusion/escape route for most of the rest!

"I have to wax my cat"

I have no idea where I got that line from, but it's stuck because of it's absurdist hilarity.

I don't say this as a serious reply of course. It's intended to communicate that the nature of the request is such that I feel I shouldn't even have to decline it. In practice usually it just generates friendly confusion, which benefits all parties.

disclaimer: I don't have a cat. Nor would I wax one if I did.

That's almost as useful as "Look! Over there! A distraction!"

Very useful indeed.

That's quite meta, if I do say so myself.

I would, but I don't want to.