Ability to see inventory space in character select screen
 

I'll admit I'm a hoarder; as such a lot of times my bank fills up to capacity, and I need to offload the items somewhere else.

Is it possible to code an addon that shows each character's current inventory space on the character select screen, so I don't keep logging into alts that are already at max capacity?

It is not. It is possible to record used space on log out of each character and show it in game on any character.

Add-ons aren't even loaded at the character select screen.

Workaround:
You could dump the inventory space to the savedvars (I bet there are already addons calculating the inventory space, bank space etc.) and check the savedvars tehn with a texteditor before login to see which char got what inventory space.