I cant speak for egg allergies, however I had a friend who visited me in Kuching earlier this year. She is coeliac and managed ok, despite having reservations beforehand. She stayed in the Ranee suites and Basaga whilst in Kuching, and also at Permai.
Ranne and Basaga were helpful - speak to the owners in advance of booking for advice and explain the situations. Then if you decide to book, try and speak to the chef whilst you are staying there. For Permai, the breakfast choice is a little limited (buffet) but what my friend did was stock up on gluten free produce in Kuching before arriving at Permai (the supermarket Everise in Plaza Merdeka stocks some gluten free snacks) Of course fruit is abundant - however MANY dishes use soy sauce, especially the chinese ones. There aren't really other eating options for breakfast near Permai. The place itself is great for kids though, situated on the edge of the south china sea and with the jungle on your doorstep - its fascinating.
The malay word 'Gandum' is useful to remember as it means 'Wheat'
Asking people for a dish with 'no Gandum' might help. The issue in cafes is that you need to make sure you speak to someone with good english and that knows what is put into the dishes - as some may be unsure of the ingredients - that's why its best to talk to the chefs if possible. Egg is used in alot of dishes, and there could be the risk of 'cross contamination' another reason to speak to the person who actually creates the food in the first place.
Kuching itself is worth at least 5 nights - and Permai at least 2. Can't vouch for eating logistics in KK, however its also interesting with the snorkelling offshore - so I would probably allow another 3 nights for there - which gives you time left over for Singapore and KL.
To sum up, i would recommend contacting accomodation in advance by email and / or phone, to establish as much as possible what they can do to help with the the allergies. Eating out in cafes/restuarants on the whole, may be a little tricky. However stocking up with snacks from supermarkets that sell imported goods (such as mentioned above in Plaza Merdeka - as well as Ta Kiong supermarket in the Spring shopping mall) could be useful.