Karloff is dumb (and needs help with breeding)

[Tad Carlucci]

The idea is to hide the Foxie ears behind something, then breed back to produce the Foxie again; only now it will be provably 'pure'

As you saw from the OP, if there is a hidden, it will eventually pop out to bite you.

Let's take a Genesis/BooBoo and cross it with Catherine.

1-in-2 will show Genesis, 1-in-2 will show Boo Boo. We don't care which.
1-in-2 will hide Foxie. 1-in-2 will hide OBB or better. We want the Foxie but don't know which boxes it's in. We need as many of these as we can get; hence using both Catherine and Nicholas to produce them.

Now we test each of those offspring to find which have Foxie. This is where OBB can help. We're looking for Foxie grandkids (great-grand of Catherine or Nicholas), but discarding them.

When we have a pair which are known to hide Foxie, we breed them together for 1-in-4 odds of proven-pure Foxie.

The complication is finding mates for Catherine and Nicholas which don't cause you to lose the other desirable traits.