My son had multiple food allergies and up to one year old our specialist's recommendation was to only have green veg, rice, potato, apple or banana!
We were in HK at the time, my specialist when we moved to the UK {when he was 10 months old} was quite surprised by taking such extreme action, so I'm not recommending this!
What I did want to say though that what I realised, was that actually although it's really frustrating as a parent being so limited, actually for the child themselves it's fine.
Kid's like repetition, until a year old, food is just for them to get an idea of texture etc, so don't worry about staying with the foods he knows, it's not dull for him I promise. :)
Anyway more practically here are some more suggestions:
Any of the 'safe' vegetables {root and green ones are the safest} make great finger food snacks. For the 'soft ones' like cucumber or pepper cut into sticks, for 'hard' ones like carrot grate them. For things you might usually cook like brocolli, lightly steam {and then cool} them to make them soft.
Base snacks around rice or oats.
So make rice balls, or oat bars that sort of thing.
Pizza 'swirls' are good since you can't have cheese. Use puff pastry, spread with tomato paste and then spinkle finely cut mushroom, pepper the usual pizza stuff, as long as he can eat it on top. Roll the pastry up and then cut into scrolls.
Cook in the oven until browned / flaky.
These freeze well, so if he likes them you can do a big batch and bring them out.
Look at other cuisines that don't use dairy for idea, such as Chinese and Indian.
Finally BLW doesn't restrict you to finger foods. Get him a long sleeved / all encompassing bib and let him eat things like risotto, and pasta with sauces, porridges, any food he can tolerate really with his fingers too. Makes a dreadful mess, but they love eating food this way, and will give you / him a lot more options.
Good luck. :)