The 403 issue is fixed in a new release of Bitlbee, but those of us relying on third-party servers running older versions need to use a workaround. Comment #3 by wilmer on the tracker ticket says the workaround is to issue:
ac tw2 set base_url http://api.twitter.com/1
However, applying this to settings on my server running Debian Lenny took some effort, so I’m making a note of the working format here:
account set 2/base_url 'http://api.twitter.com/1'
(where ‘2’ is the number of my bitlbee account connected to twitter)