The "spare tire" will cause you to arch your back more to make room for your lungs to expand; this might be why your back hurts in the drops. The guys riding in crits are typically more fit than your average bear. To get better aerodynamics (as opposed to merely increasing the number of hand positions comfortably available), you will have to get into better condition.
It is also a matter of getting yourself flexible enough to get into the position. I started racing five years ago; the position I was in then will feel like a dump truck to me now. I have extended my stem 10mm and dropped it by about 3" over the course of the last five years. But put me back to the conditioning and flexibility I had five years ago, and I would not have been able to do much in the drops on my current bikes.
FWIW, I can stay in the drops as long as I need. That is to say, well more than an hour if conditions warrant.
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Cat 2 Track, Cat 3 Road.
"If you’re new enough [to racing] that you would ask such question, then i would hazard a guess that if you just made up a workout that sounded hard to do, and did it, you’d probably get faster." --
the tiniest sprinter