How big a trailer? Is the load small enough that you could throw a sacrificial tarp in the trailer then pull it away from the pile after you dump it?
I used to dump wheelbarrow loads of manure into a 330 bushel spreader then take it out when full and spread it. The only thing I had to do was watch the weather and tarp the load if it was going to storm. That and keep the chain properly tensioned. I learned that I needed to that the hard way. Trust me, that's a lot of manure to shovel out by hand. This was in northern PA.
My spreader had a vinyl floor and galvanized sides. While a little would stick to the sides, it never stuck to that floor. Like mentioned, manure in a pile does generate heat. The vinyl floor would be your best option. You can get sheet goods from place like Tek Supply.