So there are 2 groups of people.
1. registrants (people having babies) - branding here is important and the name could bite me. Let me know if you have any suggestions for other names.
2. people buying the registrant gifts - BabyList doesn't even have search which is a default wishlist/registry feature. I think giving someone a pretty url slug and letting them email it/share it/link to it, is not going to be a problem.
If it were a wedding registry, I would suggest maybe HitchedList? It would be better than WeddingList since it's less general. Obviously this doesn't help directly, but maybe will inspire some ideas!
I think the best startup names are those that relate to a market's pathos.
My advice would be to use the proper spelling ('Babylist') in general - especially in your html titles - and in no time you'll be #1 for the term, regardless of spelling:
People will go to babylist.com. They will probably copy the concept if it takes off.
Bad idea.
It seems silly to try and compete with established players.
2. the owner of the domain is a single-location brick and mortar retailer, with no e-commerce. that would inform my thought process, were in in the OP's shoes