... does not have any effect. I see a couple of cookie entries for BCT, but I still have to rely on autofill and manual login with the checkbox filled every time I quit FF and reopen.

'Accept cookies from sites': checked.
Under the checked 'Clear history when Firefox closes' dialog box, everything with 'History' in the option and 'Cache' is checked.

Other sites remember 'always stay logged in' through browser quits, with these settings, just not BCT. Does SMF depend on cache not being cleared to Always stay logged in?

Nope. I guess it was falsely identifying as a third-party cookie, the only kind I have set to not keep on close. I just permanently explicit allowed it, restarted, and no more login required. Yay.