Under the circumstances listed, and because she's an adult, I would not list it as an allergy on your ADR. Just ask the server to bring the eggs on a separate plate.
Disney treats all "allergy" notations on the ADR (and any mention of the word 'allergy' to a server) as an anaphylaxis level allergy issue. Most restaurants and servers take the allergy, rightly, as very serious business. If you note it as an allergy, there's a good chance Disney will play it safe and only bring her the egg-free versions of everything. They usually err on the side of caution, which is really reassuring to those of us with anaphylactic reactions to a food item.
Most often, servers won't even bring water to the table until I've talked to the chef. Even at restaurants like 'Ohana that don't serve anything containing my allergen (fish) I have to wait for the chef.
I've had a server refuse to let me order off the regular menu, and insist i order off the allergy menu, even when what I wanted absolutely didn't contain my allergen (wanted a regular bun on my burger as I'm not gluten-allergic, but he'd only bring me the allergy menu burger on the gluten-free bun).
In addition to fish, I'm also allergic to certain types of fresh fruit. I only ever check off the fish allergy on the ADR, however, because the fruit allergy is self-manageable since it's only certain fruits, and only fresh (cooked in something is fine). I just don't eat the fruit items (which are easy to see, kind of like your mom with the eggs). It's way easier than having to negotiate with the server that it's okay to bring me the banana bread pudding because it's cooked and that's not a fruit I'm allergic to anyway. And, yes, I've had those fights with servers in the past, which is why I don't list the fruit as an allergen anymore.
If it was a child with the allergy, or if the allergy was to all eggs, or if it was an anaphylactic reaction, yes, it would be safer to have all the allergen-containing foods removed from the meal. But your mom is an adult, and knows what she can eat. Just asking for the eggs on a separate plate should resolve your issues, IMO.