A couple of considerations:
Do you need to be able to secure it when it is still hot/warm?
What will happen to the ashes during handling and/or going down the road (even if you tip it out, there will always be residual)?
My immediate thought was that it would cool fairly quickly once the coals are out of it, and you could rinse it out with water after tipping to reduce the ash problem. Next, I was thinking that rubber bungee cords (the black rubber ones w/o cloth wrap) would be best for strapping the lid, and perhaps the rest of it down. The trick is in finding the straps that are the correct length for the application.
The rubber should be more than adequate for the light loads and would be better at holding tension, whereas the ratchet straps, if they work loose will not tighten back up; and you won't be able to crank them down too tight to begin with w/o crushing the BBQ.
Does your BBQ have the tripod legs with wheels? I wonder if Mineralac clamps could be used to secure the base to the trailer.
Such as PN 3006T79 (or similar sized to suit) referred to in McMaster-Carr here as "Bottom Loading Clamp Hangers".
http://www.mcmaster.com/#clamping-hangers/=oq1gurI bet you can find a source there, or something similar. They are used by electricians ("sparkies") to hang rigid conduit and EMT.
Drill and tap your frame for a few of these, screw them on top with the opening side up, drop the base of your BBQ into them, install and tighten the clamp screw. Viola.
