I'm not a remodeler — my background is electrical, commercial services, and now lawn care — but I'd frame this as a "choose your battles" decision, and the deciding factor is project duration.
If this is a multi-week project, the customer's request is actually reasonable. They live there. Weeks of floor protection through the middle of their house is a real quality-of-life cost, and David's point about pricing the daily setup/teardown into it is the right mechanism — respect the request, charge for the labor it creates.
But if you're in and out in a few days? Bite the bullet, eat the few minutes a day, and move on. The goodwill is worth more than the friction, especially with a customer who's already hung up on quote language — pushing back on this becomes the thing they remember about you, not the quality of the tile work.
Either way, the lesson you already identified is the real answer: it goes in the contract going forward, with the daily-removal option priced as a line item so the customer chooses it with the cost attached instead of negotiating it after demo.
Sometimes flexibility is worth more than winning an argument. If the request is reasonable, I would offer a paid option and keep the relationship positive.
