Mozilla have rolled out Firefox 66.0.4 which contains a fix for this issue. There are still a few bugs they have to fix to get everything working for every add-on for all users.
As far as I can tell, there are no outstanding bugs that relate specifically to Kee.
From the current list of known bugs, my best guess for anyone still struggling with this problem even after upgrading to 66.0.4 would be that it relates to your use of a Firefox (NOT KeePass) master password or your file system permissions. There’s more details in this bug: https://bugzilla.mozilla.org/show_bug.cgi?id=1549249
If that doesn’t help, try the Mozilla support website: https://discourse.mozilla.org/c/add-ons