My bank just broke compatibility with my password manager (could be the plugin's fault, I suppose) and another financial institution specifically told me that their site was not compatible with password managers. They have this 'nifty' feature which turns all but the last few chars of your username into stars on blur and I suspect they just didn't want to bother to do their js event handlers properly. Fortunately, it turns out I can make it work with the right sequence of clicking inside and outside the input boxes.
Even though the login behavior on a web app's home page can cause hiccups with password manager compatibility, the alternate login form on the "You are now logged out" page is usually barebones enough to be compatible.
I used to use that method for the one that "didn't support password managers" before figuring out the click dance. Just tried with my main bank and they don't have a login form on the logout page.