If the IFA fanbase doesn't want to support an All-Ireland team, that is their decision and a decision that should be respected by the FAI fan base.
However the British Home Championship in its most original format was a competition between the four distinct nations of the then United Kingdom of Great Britain and Ireland. Those four "home" nations were England, Ireland, Scotland and Wales. If the English FA are looking to organise a tournament between the four home nations, the four footballing associations that represent those four nations should be involved. Following the partition of the island of Ireland, the IFA assumed the mantle of representing the Irish home nation in the Championship. This perhaps is linked to the fact that the IFA is the first established Irish footballing association, fielded their team under the guise of Ireland for 50 odd years after the split with the FAI, and that the IFA have fielded teams with an All-Ireland representation (even long after the establishment and regardless of the FAI).
There are five options open to the English FA as they set about organising the make-up of their tournament;
(a) run a four team competition with the Ireland team as a co-operation between the FAI and IFA.
(b) run a four team competition between the four associations that represent the four distinct nations of England, Ireland, Scotland and Wales.
(c) run a five team tournament that includes both an IFA and FAI team.
(d) run a four team tournament between the four founding associations of the tournament.
(e) realise this tournament will open up a can of worms that is best left alone.
Option (c) is the most likely outcome with option (e) running it a close second.