Alliance division between Heathrow and Gatwick may be difficult and unfeasible. But there is another way to make the division: based on destinations.

We know BA and Heathrow is strong in North America destinations. So how about:

1>, Move all unaligned airlines to Gatwick
2>, Move all routes to and from Far East, Australasia, Middle East and Africa to Gatwick
3>, Keep all traffic to North/South America in Heathrow while maintain a large European business destination network for feeder flights
4>, Grow some European business destinations and keep leisure routes in Gatwick in order to feed the Far East/Australasia/MEA routes

Traffic from Far East, Australasia, Middle East and Africa is likely to end their travel in Europe, and majority domestic. Thus separate them from north America should have no problem. Although there are traffic to Americas from Middle East and Africa but that would be up to airlines to plan to argue their cases to stay in Heathrow.

Keep North American (and South America) traffic in Heathrow would let all three alliance still stay in Heathrow with a pan-European network.

The biggest problem would be Emirates. They may stay in Heathrow but likely to be objected by other airlines that are forced to move without being in an alliance...