Same problem here.
Finish first quest with Zed as Guest on my friend game.
Game crash just before the end. We continued it and finished it.
Quest is completed and everything unlock (seconds quest etc.) but not the first slot of chip.
Zed is lock like this both in my local game and on my friend one.
Was playing coop, disconnected once because of a game update. (The game crashed btw)
After beating the run, the 600 scraps quest was completed, but Zed's 3rd slot didn't unlock.
Might as well check other heroes, they can have the same issue.
P.s. As I mentioned the game update, this bug happens between ver 1.1.10083 and 1.1.10087 - Latest version.
Hope my progression gets fixed in the future.