Assuming there weren't already problems with the balloon idea, what you'd have then is a pressurized vessel, the pump, and of all the compressed helium weighing you down for 99% of your trip just to save a tiny bit of energy (and I'm assuming a day with zero wind here) during the offloading procedure which probably doesn't even last long enough for the balloon to fully inflate.

Also, as far as I know, the cheapest/lightest material to make the balloon out of would be mylar, which is conductive and not something you want around powerlines.

You'll experience the same scaling issues with whatever it is you expect to inflate with. You're going to have a lot of volume dedicated to the pressure vessel and not so much in the pressure vessel, because of scaling issues.

And I'm discarding the pump to put it back, assuming we're just going to vent to the atmosphere when we're done. Things get worse if you want to pump it back in.

No that doesn't really work. High pressure helium pumps are expensive, heavy, and require a lot of maintenance. Plus they require a lot of power to operate. And they produce a lot of waste heat.

Inflate it with what? Raw air? That would do nothing.

Are you proposing that it could somehow hook itself up to a helium or hydrogen source, while hovering?

Not saying this idea is worth the complexity (sounds over-engineered), but I am pretty sure you could make like a 1"x2" can of compressed helium attached to a 3' balloon with a simple valve to pop open and inflate the balloon. Think of those little air cans that you use for refilling bike tires on the go. Only weigh a few grams, holds a lot of air.