None of them sounds right to my BE ear. We don't say 'check out at the cashier'.
We say 'pay the cashier' or 'pay at the till' where I live.
But another BrE term for the cashiers/till is check-out. I use it as a noun, but not as a verb in the supermarket context.
You don't "clear the bill". In a place where a bill exists, you pay it. That doesn't happen at a supermarket though. It happens at a bar or a restaurant, or any place where you could be presented with a statement of expenses you have to settle.I clear the bill at the check-out in the supermarket.
In the case of a bar or a restaurant, you could say "pick up the tab/check".
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