I think we might be talking about different things.
The site where the Apple password is not working has accepted autofill from Last Pass previously. I recently switched from a Windows laptop to a Mac and found Last Pass to be much less functional (at least on Safari), so decided to start using the Apple passwords app. For the sites where I have implemented, it seems to work well. But on the problem site, the login page remains empty, even though the app is open and I can see the entry for that site.
Alternatives I will also explore: using Firefox - or Chrome - (and hoping that the Last Pass extension works there) or transferring everything to Passwords.
I also just noted that even though I am not logged into Last Pass, the user ID and password fields on the problem site still have a grayed icon for Last Pass in place. So maybe the site, or Safari, are retaining something.
I did look at the tool you linked to. It is beyond my current level of knowledge.
(I actually - obviously foolishly - switched to a Mac thinking it would be a great deal closer to the iPhone and iPad experience than it is, since I have been using both for many years.