Pay with different CC
 

I have a non-prepaid hotel reservation booked by my customer.
Will the front desk let me change the cc on file when I check in or will I be forced to use their cc?

You can call the property to double check, but this shouldn't be a problem at all. Pretty standard for them to require a physical CC at checkin and they're happy to charge whatever you give them. The CC on file is mainly to hold the reservation, not necessarily to settle incurred charges.

I concur. With work, my travel department holds the room with the corporate card, but the expectation is that every employee pays for the hotel stay with a personal card and is reimbursed after the fact. Totally not an issue at all (and as @ClarkGriswold says - they always will ask for a card and will typically charge to the card presented).