If Thrifty state that you need both parts of your driving licence then that is a condition of the rental. The paper part carries a record of any any driving penalties. You could of course risk only arriving with the plastic laminated card but Thrifty would be within their rights to refuse the rental (as you point out it could be down to whoever is at the desk when you arrive). If they refuse the rental your options will be to find another car rental company who will accept just half of the licence and pay whatever the applicable walk-up rental price is at that time (often much more than a pre-booked rate) or accept that you don't have a car for the holiday and use taxi's and public transport.

I normally hire from Dollar (which is part of the Dollar Thrifty group) and have experiences of both them requiring both parts of the licence and just the photo ID part. As I have hired cars from them often, they seem to have my details in their computer and over the past couple of years have just looked at the photo ID part and asked if I still reside at the same address.

I think you are wise to try and contact the DVLC and seek a replacement or copy. It will save you the worry of just taking the gamble.

Also, you could see if Thrifty have a loyaty programme or advance registration option on their web site. Dollar and others have these and you can input your details in advance and the idea is that you save time upon collection and the data is already in their system. With Alamo you can do this and even walk straight to the car lot and jump in a car. If this pre-registration option is available you might just find that by pre-keying the data before getting to Orlando the rental agent at the other end might not question the missing paper part of the licence. No guarantee of course but another option.