I understand the excitement! we too got a green light for baked egg just a few weeks ago. my son tolerated baked egg without incident in cakes and breads but our allergist took it off the table last year since he tested positive and he does react to straight up egg. fast forward year later with new allergist he says if it’s tolerated then we need to eat it.
i think coconut milk will be just fine. i’m not quite sure the exact science behind what the milk protein provides in terms of cake structure. i feel like milk may provide some tenderness to the crumb (BIG guess here). i know that from making wacky cake for my son all year the tenderness can be achieved w/out milk too. i once used coconut milk and a tablespoon of vinegar like the wacky cake and it worked very well.
if i were you i’d experiment a little if you can. cut a recipe in half so you don’t waste too much and try soy milk, coconut milk etc.