I would go to a tent and awning shop and have them sew up a nylon webbing "basket" that fits the base and goes underneath and around just under the lips near the top. Then a loop handle that clears the lid when you open it. Or being from Texas you could have a tooled leather sheaf with a handle made for it. Now that would be cool and unique. Kind of like Waylon's guitar.
My XL was steady as a rock while on a 24" paver in the table. My medium in a table nest seems to be a bit less so. Will let you know about the XL and table nest in the next couple of days as I have the table stained and the neighbor lads will help me put it back together soon. It's handy to have a couple of high school athletes nearby. All you have to do is promise them pulled pork and just about any little job is done in a flash.