After consistently seeing the extra movement speed buff from the Archive go away in between rounds, I have come to a conclusion about what's causing the bug.
-Werewolf Behemoth avatar transformation: This removes the speed buff 100% of the time. The only way to get the buff back is to log out.
-The puzzle side room (the one with the switches): There's a very high chance that this room bugs out the speed of your character inside the Archive (I'd say about 75% of the time).
-Tho'at fights: Sometimes, none of the above caused the speed loss, which left me and my friend to think that the Tho'at fight at the end of each Arc could be the perpetrator of the bug. This one is not confirmed, though.
The loss of speed is VERY noticeable. So this is NOT a case of "guessing whether we had the buff or not" (since there is no way of actually seeing if you had the Archive speed buff active). As I said before, relogging solves the issue, but it is a huge time sink to do it everytime the bug shows up. It's also a run-ending bug, since if you happen to forget to relog before important rounds, you might as well get killed due to being so slow.
On a side note: I'm a Werewolf, and I've experienced this bug more often than my friend who duos with me constantly. Maybe there is some kind of code shared between the Werewolf Behemoth and the normal Werewolf transformation that makes the speed buff not work?
On a second side note: I suspect the speed loss is related to the Infinite Archive speed buff granted by one of the upgrades (unlocked via achievements) you buy at the merchant in there. I'm fairly sure this is the only hit on my speed I'm getting.