You can, but not the way everything is now... but you can 'cheat'. Box in 6-8" sections at a time, and stuff the backside of the plate, and around the fuel lines and wires, with a welding blanket... then just use the blanket for each section you're welding in(don't leave the welding blanket stuffed in the frame, obviously). It's slow, monotonous, but easier than pulling all the wiring and lines, boxing in the frame, and remounting everything. The problem with boxing it in sections like this is, IF you have a fuel, or wiring problem in the frame, you'd be screwed. You also have to be careful about any mounting bolts, so that you don't box those in, and :hump yourself there, too
The easiest way to solve your problem is, to get a few more stock crossmembers from junked pickups, and mount those to yours. I would bolt those in tho, don't weld them in
Ok, well now that I have my 4wd swap done, I have a crossmember that I can use to stiffen up the frame. Now my question is where would be the best place to put it, and should it go on top or bottom rail? I'm thinking that somewhere near the front of the bed/back of the cab would be the best?