Just FWIW.
A sideload works (by debugging), however no real solution. 
Therefor for you and me (and more users) no solution. You are right!
As for my situation, 2023.8.3 has solved the PIN issue. Can’t say it does for everyone (see FunkyB).
On the other hand, I messed up the log-in-by-PIN-settings before, trying to solve things, so I had to correct them first.
I understand your frustration, but Bitwarden has corrected the issue. It’s up to Mozilla now to release it.
They seem to move at a slow pace.
Well, it would seem I have gotten the attention of at least two folks on this thread, but my point still stands. Taking a month (or technically 3 weeks per Go12’s post 7 days ago) to identify a problem and then supposedly issue a fix still feels like an eternity for people who pay for the use of this program and who swear by it and tell others to use it also. At this point, I really do not care if my tone and/or words have come across as rude or out-of-line. I, as a paying customer of a product and service, expect better and would appreciate the people who develop this open-source product to be more transparent. Furthermore, if there is a problem with Mozilla releasing updates, then let us know that and tell us who we can go to and complain about their broken eco-system. In the end, I just want what I am paying for to work properly and not be told to jump through this hoop or perform this action as a work-around. I personally feel Bitwarden is far better than that and that is precisely why I left another company who was not any of what I mentioned above.
Actually, there are two separate issues that have been discussed in this thread. OP’s post is in regards to Bug 6169: Biometrics Not Working On Firefox, which was first reported on September 1, and was caused in part by a mismatch between the Desktop app and Firefox extension version number when changes to biometrics functionality had been introduced in version 2023.8.2.
Later, when version 2023.8.3 was released, this introduced Bug 6286: Firefox Extension — Can’t unlock vault with PIN, which was first reported on September 14, and patched on September 20.
In both cases, delays in making Firefox users whole have been caused by the slow review process for extension releases by Mozilla.
Not a bad answer but passing the culprit (again) ? 
The question remains: Should Bitwarden as a serious Password Manager not take up its responsibility and urge Mozilla to hurry up?
Isn’t Bitwarden in essence for safety on the internet, and not for a fancy-look?
See also : PIN Unlock in Forefox no longer working in 2023.8.3 - #25 by sjokoboy
Probably pure coincidence 
, FF add on is now updated to 2023.9.1 
@sjokoboy
It sounds like Bitwarden did “urge Mozilla to hurry up”:
I am now running FF browser extension 2023.9.1 and the PIN unlock feature is working correctly again. I am not going to get into a finger-pointing war of who did or didn’t do what, but suffice it to say that problems like this should be dealt with at a much faster pace going forward, especially when it deals with the core functionality of the software vs. a cosmetic change or nice-to-have. And yes, I do understand this thread dealt with two separate problems, however, the PIN problem lasted for approximately 19 days itself (and yes, more than likely due to FF changing its codebase). I also requested greater transparency as well, not just - we are aware of the problem and are working it. Moving on.