Thats a good idea for your reservoir!
I can’t see a problem with keeping it inside the tent, I’ve a feed bin inside my tent that fills from the reservoir, the idea is the nutrient gets a chance to heat up between feeds.
for my autopot I use that 10ltr tub but i only put 4/5ltr in it every couple of days that also stays in the tent for the nutrient temp.
The only thing I can think of is keep an eye on the nutrient temperatures during summer.