Getting new passwords for the same site is exactly what the counter is for. Anything you would do to get a "new" password has to make some kind of a change to the site configuration.
I do see your frustration, though. Website password policies force us to do annoying things like these. Restrictive password policies have been contra-indicated for a long time now, but many websites keep using them for lack of a better understanding of security.
As the years go by, website owners are beginning to realize it is better to allow for simple but long passwords than to force a human to pick characters from restrictive sets of values. Hopefully we can all transition to a default counter 1 Maximum Security password for each of our sites in the next few years as site owners fix their password policies.
Actually it would be nice if we could use a shared file for our site and iteration informations (let's say I'd have that in my private cloud storage). So no matter what device I'd use (as long as it is mine), I'd have the latest information and therefor the correct password available. If the app would sync it locally that would be even better!
In our company I'm forced to change my passwords regularly, Often enough that I don't even remember which iteration I'm in with those passwords. Which makes it nigh to impossible to get the right one if I had to use another phone ...
So a common store for one's devices would be very much appreciated indeed.