I found an egg substitute yesterday in our local supermarket health section. Its a local product called egg-like. I'd be very surprised if you guys don't have a number of these type of products. The ingredients are : potato starch, tapoica starch, vegetable gum (461), mineral salt (170,329), food acid (330).
You mix a heaped teaspoon with 30g of water to make up substitute for 1 egg. I used it to coat fish before dipping in polenta and it worked really well.
Could be worth a try if you can tolerate the ingredients. I'm really sorry I wasn't much help. I do hope this helps.
I also noticed there was oat milk there as well. We Aussies are told not to have oats (because of contamination with wheat in the processing). I notice the American Coliac Society says they are all right. You guys probably have better controls. Perhaps you could check out oat milk with your local coliac society?