By BOX TRUCK, I'm guessing you mean "cab cutaway" which is a van front end and box on the rear.
We keep a couple box trucks around for emergency deliveries, most of them have load capacities of 2500 to 4000 pounds (assuming dual rear wheels) with GVWs somewhere in the 12,500# pound range. I think the weight capacity would handle the BX, but you might run into length issues. Those trucks typically have about a 14' long cargo box, I'm not sure what the L.O.A. is for a BX with FEL & BH.
The biggest problem you might have, given the tight quarters of the box will be centering the load so that it doesn't put too much, or too little weight on the front end.