Boy, things must have changed since I was in the military (20 years ago), we could pretty much carry what we wanted... Sorry to hear of the PC crap.
I have and love my TM, but don't have a BR yet. I do, however, have a CT and RS (sorry for the abbrevs, but I'm tired this am), and can compare them.
I like them both. They both have enough forward weighting to be good choppers, which is something that I look for in a larger knife. They both are easy to get and keep a sharp edge, but the CT responds to steeling better than the RS, and also the convex edge will be stronger and more resistant to damage from use/abuse. Also, the customer service from Swamp Rats is FAR better than Cold Steel. The SR101 in the CT will be more abrasion resistant, and should be a superior steel to Carbon V, although I haven't been able to test it enough myself to say that it is, I'm merely repeating what others have found, and I believe them.
All in all, the CT is a better knife, for the same price range. And don't get me wrong, I really like my CS blades, but the facts are the facts. I also really like my CT. I have medium hands, and like BOTH handles, although the Swamp Rat handles fit my hands a little better, and are more comfortable in hard use. They are probably more resistant to chemicals, DEET comes immediately to mind.
The BR and the TM will be much better choppers due to the extra length and weighting. There's no getting around that. They are also much more awkward to carry, no getting around that either. The steel and edge comparisons will probably be the same as the RS/CT.
Your best bet would probably be a Leatherman or similar tool, and perhaps a Howling Rat. If you are allowed a CT or BR, great, but it would be heartbreaking to have it confiscated. I'm sure once you get to your final unit, Swamp Rat will be happy to send you whichever you want.