Yes...mount to the frame.
Decide if you want to step onto the trailer to cook or want it to be ground accessible as both have advantages. It's hard to cut up new trailer but remember you can cut the floor/notch the crossmembers to drop the firebox thru so you can adjust the height and foe/aft location. With weight on one side, you may need to change the springs out to balance the trailer but remember some weight will be on the other side with coolers, wood, tables, etc.