I would like xdg-open to use sensible-browser instead of the default
firefox-esr.desktop. But I get the following error:

qaa:~> xdg-settings set default-web-browser sensible-browser.desktop
xdg-settings: $BROWSER is set and can't be changed with xdg-settings

$BROWSER is the configuration for sensible-browser, so I don't
understand why I get that.

Hmm... the xdg-settings seems to use $BROWSER internally. I don't know
the intent. In any case, $BROWSER is not honored by xdg-open (which
uses firefox-esr.desktop instead of $BROWSER).