How to go about quickly doing the Training Tower Quests

There's not really any way around the stamina costs. Focus only on the quests that grant orbs, and never lose a unit. Check your opponents before entering the map so that you can just throw out the draw if you see some enemy that's going to be a headache (some cavalry mage or super-tank, for example). Try to play periodically throughout the day so that your stamina recharges on its own.

Aren't there Oct quests that give you more stamina potions? 🤔 If you're willing to sacrifice at the very least 1 potion, later on you could replenish it though the quests. I think one quest gives up to 3 but I'm not 100% sure. 👌

The best way to do it quickly is to have a strong team for each type clear the last round. Assuming you mean the final 4 quests. Stamina costs mean you just have to use all of your natural stamina, then wait until its back up to do it again,

If you have the teams that can consistently clear the Stratums without any casualties in order to save yourself from wasting the Stamina. I'd also consider using your naturally gained Stamina before blowing any pots, only using them if you feel like you're on a time crunch for the banner.