Support Gem Capacity not working with Weapon Set Passive Skill split stats

I'm playing Sorceress and have an Ice Staff in Weapon Set I and a Lightning Staff in Weapon Set II.

My current Dex is 12 and I have a Dex gem on Freezing Shards, which is bound to the Ice Staff in Weapon Set I, and another Dex gem on Frostbolt, which I set to only use in Weapon Set I. Which puts me at Dex (2/2) for the Support Gem Capacity.

I split my Passive Skill Points so that Weapon Set I would get Breath of Ice, which gives +10 Int, and Weapon Set II would get Breath of Lightning, which gives +10 Dex.

When I have the Lightning Staff equipped, my Dex goes to 22 and my Support Gem Capacity goes to Dex (2/4). I then added another Dex gem on Orb of Storms, which I set to only use in Weapon Set II.

So I have 2 Dex gems in Weapon Set I, with 12 Dex stats, and 1 Dex gem in Weapon Set II, with 22 Dex stats. My assumption was that Weapon Set I would use the first 10 Dex points to cover its 2 Dex gems and then Weapon Set II would use its additional Dex points to cover its 1 Dex gem, but when I go back to the game Frostbolt was immediately unbound and not in the bind list anymore.

I know from playing Witch before that there's a menu for setting specific Spirit reserve spells for each Weapon Set. So I'm wondering if there's a menu like that for Support Gems that I'm just completely missing, or if this is bugged, or just not how Weapon Set Passives are supposed to work.

I also noticed that weapon switching doesn't automatically update the Support Gem Capacity values. When I'm on Weapon Set I, my stats are Dex 12, Int 76 and Capacity is Dex (3/2), Int (6/15). When I switch to Weapon Set II, my stats are now Dex 22, Int 66, but my Capacity still shows Dex (3/2), Int (6/15). If I do 2 unique spells using the switched Weapon Set, the Capacity will then correctly show Dex (3/4), Int (6/13). I'm not sure if that's directly related since even after fixing the Capacity values, removing the Dex gem from Frostbolt, rebinding it, and adding the gem back, Frostbolt still automatically unbinds itself.