I don't know.
I've already asked this question on another server about a month ago (in the help centre) but there's still no reply. =(
I'm very curious for the answer too though.

Eggs can only be applied when there's no covering left, so it might use the egg. On the other hand the VIP perk always is the 4th covering. I'd hope it'd take the VIP covering and keep the egg, but I could also imagine they didn't think of that specific case.

Sorry for raising hope because you got an answer and still being no help at all ^^
I just wanted to say: if you find out but the topic is closed somehow before - please tell me ^^
I occasionally have VIP on the other server due to Kirin, but I use Eggs for daily objectives every now and then. It'd be a waste if they just disappeared, so I got curious.

It's ok, while I was drafting the question I had a suspicion it might be one of those situations that not many people have come across so it might be difficult to find an answer without testing it myself. I was tempted to just put it back in the sales in the hope that someone without the Tianma perk would end up with it, but at the same time I'd really like to know what would happen as if someone who does have the perk bought it I'd feel guilty for just passing the issue along to someone else.

I'll leave it a few days to see if anyone has a definitive answer and if not I'll test it out and report back.

My curiosity hopefully will have overcome my reluctance to potentially waste a BMI by that point

Bebi did you breed the donkey yet?

Thanks for the reminder, I'd forgotten I'd been meaning to do this

I've just added an Egg to one of my female donkeys and tried it, the male donkey in question used his Egg and now says 4/4 coverings, so it will definitely use the Egg in this situation.

Thanks Bebi, I'd been watching this too So I'm guessing it stays at a max of 4 rather than the 5? If that's unknown I might try it with a draught uni and use a hestia to check.

That's how I understood this:

Careful, Tianmas VIP perk only works for donkeys! So you shouldn't even be able to be in that situation with any draught horse.

I went and checked, I was clearly having senior moment as your're right