While you're playing with wine's default registry, could you create a registry entry for https? It can be just the same as the http rule, e.g.
HKEY_CLASSES_ROOT/https/shell/open/command = winebrowser %1 (or whatever the http rule ends up being.
I have an app that assumes that entry exists and it exists in my windows XP registry, tho I am not sure who created it.
Thanks... mo