But then you have to remember the counter setting for that website. So it's almost the same problem; I could just add a 1 to the end of my master password for that website.
I've just installed it and haven't yet used it, but as far as I can tell from the docs, the counter is stored as part of the syncable profile for a site.
The profile is also used to store other site-specific attributes, like if it has some weird password rule that requires pure alphanumeric or a specific password length.
No, the counter would be saved by LessPass. Every time you look up the password for that site, it'll already know what the counter is set to for that site.
This blog post[0] shows how LessPass lets you change passwords per site, and get around tricky password requirements.
The profile is also used to store other site-specific attributes, like if it has some weird password rule that requires pure alphanumeric or a specific password length.