I don't think either the Summer or Winter Courts could join nor be permitted to join due to balance.
If you permit just Summer to join, Winter would most likely need to oppose and vice-versa if you permitted Winter to join instead.
This is a valid point! OTOH, it's not like the GC is some cosmic-scale tipping-point...
It's a secret, largely political/investigatory sub-group of WC wizards (mostly; as noted, there may be other entities) trying to figure out the infiltration/betrayal of the WC by this "BC" or "Circle" or whoever they are -- and THAT group (the BC) in turn may be a group similarly centered around dissatisfied WC wizards.
In terms of Summer -v- Winter, it may look like a school-yard argument, as seen by business-executives in the office-building across the street.
The FQ's may not
care about the "balance" involved in GC membership, because there's no effect.
Of course, that analogy also leaves them with a can't-be-bothered-to-join perspective, even if they got invited. Why would a business executive join a schoolyard clique???
That leaves an option of both joining but we haven't seen much cooperation ever out of them. In fact, they almost need to be opposed, per information we've received through the story. Mab exists to defend the gates. Titania exists to protect us from Mab. That Winter grossly outguns Summer but Titania can drag Mab down with her.
I wouldn't dismiss "dual" membership so readily.
First: recall that the most-potent Queens, the Mothers, apparently co-habit readily. I think Summer and Winter have shared interests (it remains to be seen whether the GC overlaps with such interests).
Second: even if they'd
chafe, they wouldn't have to
meet -- as I understand it, the GC has a "cell" structure, where GC members mostly only know other GC members in their own cell, and only Eb knows everyone (or maybe not even him). But there could be one cell with Summer and a different cell with Winter, with those 2 cells never meeting together...
