The 4R70W would be a way better transmission than the C5 gearing wise (lower 1st gear and a high overdrive that will compensate for deep axle gears). I can understand running the C5 though if it won't be driven on the highway and you want to keep the BW1350/1354. The RBV Borg Warner cases are pretty stout for their size and will have no issue holding up to what you describe. The BW 1354 is a bit stronger than the BW 1350 and you don't have to worry about slip yoke versions. Either will hold up to more than your axles will.
As someone with a BW 1356 in a Bronco II, they don't really fit well and require some hacking to get everything to work. For what you are talking about, I'd run a BW 1354.
As for your original question, yes you can use the Explorer motor's EDIS and OBD II computer and harness without the 4R70W pretty easily; the computer will just need to be flashed for it. If you are considering running the 4R70W, the Explorer harness and computer is a pretty slick way to do it. The Explorer 5.0L swap is super popular on ClassicBroncos.com right now. There is a guy who goes by EFI Guy who specializes in the Explorer harness work and PCM flashing if you don't want to do it yourself, but he has posted a bunch of DIY information on Classic Broncos and Youtube if you want to tackle it yourself. I'd give you links, but I am posting from my phone and it is a pain in the ass to do. It should be pretty easy to find, but let me know if you are having trouble finding the information.