I understand the technical limitations. How about just adding a private stash per character? That way you never load the other character tabs, so the data update/poll issue goes away.
I’m willing to bet that this is trivial, and you can reduce the max shared tabs by 4-5 and reduce the server data request per join.(since with an extra tab per character that would effectively be 14 more)
And this fits the game, since you wish to store alternative builds or sets.
Another QOL feature that greatly matches this is to have the armory always use the character private stash to equip and store the sets in question.
And beyond this, I would settle with just having the Armory store the sets in itself. If you do this, I’d be ok, if you reduced the max tabs by half as well.
As it stands We can’t store all sets/builds unless the character is also carrying 1-2 sets on him. This is really bad!
Alternatively add more character slots!!