My understanding of how this works:
When one lifts heavy weights in the gym, one is doing anaerobic exercise, even though one's HR might not get out of zone 2. This is because even a 1 minute set is brief in terms of aerobic conditioning, and because most of the work is usually done in the concentric phase of the rep.

Taking this to bicycling, climbing in a big gear is somewhat like weight lifting. You are using full muscle power at a low cadence. So I think you are burning glycogen faster than someone climbing in a more appropriate gear, because your muscles are contracting closer to the anaerobic limit than those of a spinner. Your HR is already close if not at your lactate threshold/aerobic limit, so that means your legs are really working hard. Most authorities think we should limit our low cadence climbing drills to zone 3 to make certain that we are working our legs inside our aerobic limit.

My personal advice is to go to a triple with a 27 cassette. The added weight is a joke, and the convenience is awesome. And if you get strong enough to climb in the middle ring, good on you. Meanwhile keep that climbing cadence in the 75-85 range. You will notice that your HR will probably be a little higher for the same speed going uphill, but you'll be able to keep it up a lot longer.

Lots of good climbing in Tennessee.