Gas consumption has always been an issue for me. When I first started, I could barely last 20 minutes on a steel 120. Now I am about 45 minutes. Part of it was realizing I was overweight by about 20 lbs.
What is the advantages of SM? What are the disadvantages?
Locally there is no where to dive SM.
going backwards, and temporarily ignoring the discussion about reg swapping.
1. You live right next to a great lake, your last sentence is not accurate. You can dive sidemount pretty much anywhere you can dive any other configuration, so wherever you are diving singles now, which is apparently quite frequently, you can dive sidemount. Some boat operators don't like it, but they are warming up to the idea. If you are shore diving, you can dive sidemount.
2. Pros
Sidemount changes your profile in the cave from square to flat. This is advantageous in most caves because they are generally short and wide vs. tall and skinny.
You can go anywhere in sidemount you can go in backmount, you can not go everywhere in backmount that you can in sidemount
It moves your tanks to your sides where you can see any failures
Cons
The tanks are not manifolded so if you have a regulator failure, you lose access to the gas in that tank
It is generally more difficult to get "settled" in sidemount than it is in backmount, bit steeper learning curve
3. Overweight is bad, don't do that, but that does not explain your air consumption. You do not mention depth, so saying you can last 45 minutes on a 120 is irrelevant. A single 120 will last me about 45 minutes, and I have a very good air consumption rate, provided I'm at 100ft. So if you're diving at 100ft, you're fine, if you're diving at 40ft, not so much.
Side notes:
Sidemount regulator switching has a few "switch" protocols. The one I follow which makes the most sense to me. Since you're going to Mexico, this is with AL80's, and assume starting at 3000psi.
Long hose start, breathe down to 2000psi-start on long, gives you the long hose in your mouth for donation early in the dive.
Switch at 2000psi to short hose and breathe to 2000psi. Dive is now over and you are turning around to come back.
When short hose hits 1000psi, switch to long hose. Stay on long hose to end of dive since this is when buddy may be at higher risk of running out of air.
Total of 3 regulator switches when using al80's. If I was doing big steels filled to 3600, I breathe R600, L1200, R600-turn, R600, L1200, R600. Total of 5 regulator switches, but that's a long dive, typically 5 over the course of several hours. Not what I would call "frequent"
If you are only cavern certified, going to doubles or sidemount does not give you access to more "gas" as your TDI Cavern Diver certification limits you to 1/3's of a single tank, or 1/6's of doubles. That is the same volume so going to one of these two systems does nothing to you in terms of your bottom time, and will likely decrease it because you have to work harder to move the extra mass of gear around. You do not get access to the full "thirds of doubles" with TDI until Full Cave which you are certainly not ready for.
Given your military background, the explanation of what you are trying to do, discussion on your SAC rate changing quite radically, I think instead of looking at an intro to cave course, or a doubles/sidemount course, what you are really going to benefit from is something like a NAUI Intro to Tech, GUE Fundamentals Rec, or whatever UTD calls their equivalent. That course is going to teach you how to move efficiently through the water, help get your anxiety down over reg switching *which is critical to really cave dive*, and from there you will gain a lot more confidence and reap a lot more enjoyment out of your existing cavern certification. Bob Sherwood is about a 9 hour drive from you to New York and he will be able to help you a lot more than trying to go to Mexico and figure it out down there. His background is with GUE so it will be a backmount focus, he's not really the guy to go to for sidemount, but at this point you need to be looking at solutions to help you relax in the water so your air consumption goes to a reasonable level instead of focusing on how to take more gas to extend your bottom time. Being more efficient will not only make your dives more enjoyable for you, but also for your buddy all while increasing safety.