Once you get to that page (where payment is pending) it's locked in, there's no more fluctuation.

I get your point, some people would want to pay a $100 flight with lightning. Mine currently just happens to be $4k.

To answer your question, they use BTCPay server, but no I don't believe lightning is enabled. I think on their end it eliminates their end to maintain inbound liquidity though... They'd need some fat channels