As you've already learned from the comments, the hard part of getting to space isn't the height, it's the speed. The Saturn V broke the sound barrier less than 30 seconds after launch. So, how is your refueling plane even going to keep up? The only possible answer, at least with modern technology, is to make it a rocket itself.
So, now you've got two rockets ...
Per this paper:
Fuel consumption increases (approximately) linearly with aircraft weight, because weight corresponds to required thrust and thrust corresponds to fuel consumption.
The paper lists other fuel consumption factors as well. But, as you suspected, aircraft weight is a dominant factor.