The AISC steel code has provisions for web crippling and buckling where you have isolated loads acting on the flange of a wide-flange beam. Logically, they should have similar requirements for rectangular tubes, but I can't get the current AISC code to download, so can't check it myself.
You can check the vertical legs by treating them as columns, check the horizontal part by treating as a beam spanning between the two sides. Both of those would be fairly approximate. Including the effect of corner radius would make it more accurate.
My guess: The primary mode of failure would be sidesway due to unintended lateral loading when you put the load on or off.
The attached sketch is how I interpret your diagram- if I'm missing it, please correct.
Could you estimate the safe load by considering the shear capacity of a box section the same size as the short side of the rectangular box? The failure mode is pretty much the same and I don't think the longer length would make a significant difference...?
