Never done this on a bike trailer, but my wife and I have driven multiple thousands of miles with a couple 30-ish gallon totes strapped into a hitch mount basket on her little Kia Soul.
For my money, i prefer the 27 gallon tough totes from home depot since they are more square sided (better space utilization) and seem sturdier in compression (good for stacking or tying down with a ratchet strap)
If I was going to do that long term or with anything valuable inside, I might look into DIYing some weatherstripping around the rim to help keep water from getting inside.