[... I went ahead and did the ancient-weapons-in-the-under-the-lake-temple quest for the Dozens, but that didn't lead to anything happening on the Dozens' side of things either. They don't offer any other quests when you go back to them with the bits of the ancient weapon...]
If you've completed Bronze Beneath the Lake, and spoken with Lady Webb in Hadret's House (Brackenbury) then you should have a quest to gain an invitation from the factions (check with Lady Webb about this, in case of doubt). If so, the Dozens will give you their support since you've helped them resolve Bronze Beneath the Lake (you can ask them about it if Webb has told you that you need support to attend the hearings at the Ducal Palace).
I did the Bronze quest(since it looked like an interesting one), but I lied to the Dozen about not knowing how the weapons work, essentially thwarting them. I actually wanted to give the weapons to the knights instead, but there wasn't any such option. I don't have an invitation quest yet (presumably because I have only investigated 2 out of 3 visions), but the knights refuse to allow me to proceed with their quest line, so it looks like eventually I'll face the same problem (and I really don't want to help the Dozens). Will 1.04 address this? (and if so, when? like a super-rough estimate?)