As far as I can see, the only democratic way to deal with this now is the ballot.
The people who don't want recognition, either turn up and vote no or don't turn up. Those that do, then turn up and vote yes.
Its the only way that I can see a majority opinion at each base winning now.
If FR are confident that the pilots are happy with the current direct negotiation system, they should have nothing to fear. If BALPA think that enough people want recognition then they have nothing to fear.
Edited to say that either way I think the pilots are pretty screwed. If BALPA win then the next period will be fairly grim as FR follow through on their threats. If they don't, then when winter comes, FR have carte blanche to make cuts where they like. Either way it seems to me they lose.