I've talked to the Captain of the Watch, and she gave me permission to enter the Blacklake District, and it shows up in my Journal. However, the guards at the gate keep telling me I need to permission. I've searched around online, and maybe I'm just blind, but I can't find any other topics on this particular bug. Any help would be greatly appreciated, thank you!

I've never seen this one before. Try leaving through a different gate and see if it's on the overland map.

I had a lot of trouble with the watchman quest in two places prior to where you are.
The first time I got past it by going back to an earlier save & redoing it.
But the second broken questline was still there after going back & redoing it, so I uninstalled the game redownloaded it & started a new game & got right through to Act two with only one minor bug.

I reinstalled, but my problem persisted. I guess I'm going to have to start over again. Ugh, this is almost as bad as the time when my Baldur's Gate save disappeared.

I take it you aren't getting the dialog option with the guards that says something like: "I have a pass from the commander."?

Correct, it's super annoying that I have to start over again.