Upvote:1
A US permanent resident does not need a UK transit visa in any case, provided that she can show her green card, so your wife does not need a visa for this trip.
There is also a case to be made that the UK must allow her to transit under the EU free movement regime, because she is traveling to join you. But the EEA family permit is not issued for that purpose, and there does not seem to be anything specific in the UK regulations that implement the directive that would apply to such a situation, so I would avoid invoking the right of free movement except as a last resort.