Have you tried just eating the egg yolk? The sensitivity may be to the proteins in the white?
The other suggestion I've read is complete and total elimination for six months of the eggs, and then a very slow re-introduction starting with just a few yolks. A few a week, then a few whole eggs, never many of them, but sometimes sensitivities can be reversed.

Trader Joes has both a ghee and coconut cream at a good price.