Until another member is added to the group, there is no third member; thus "the third member" would be premature. Only once the group contains three members can one refer to "the third member".
Compare: I'd like to add a third coat of varnish. One could say "I'd like to add the third coat of varnish" if one had been planned from the get-go. But an ad hoc third coat would be called "a" third coat.
On that logic, if it had been stipulated somewhere that the group must be comprised of three members, then one could say "he was asked to become the third member."