I made a butter from flaxseed oil, olive oil and coconut oil. I can't remember where I read it. I can't seem to find it anywhere. So, I don't know what the proportions are. You will have to experiment. I don't think you have to use olive oil if you don't want to. It's the coconut oil that goes hard when refrigerated and when mixed with other oils makes a soft butter.
There is a non dairy no artificial falvours/preservatices etc butter in australia called Nutalex (made from oils) that you can mix the flaxseed oil in to. That is not bad either.
I also have this recipe but have not made it.
NON DAIRY MARGARINE
200mls carrot juice
2 Dsp lecithin granules
1 tsp salt
Heat the carrot juice and add the other of the first ingredients. Stir constantly as the mix comes to the simmer.
Melt the coconut oil or palm oil slowly and add the other oils. Add this to the carrot mix. Blend for a few minutes until very even then pour into a cooling bowl. There will probably be some separation on cooling. Allow to cool and place in the fridge. When it has set completely, mix until the seperation is re-suspended, then transfer to small containers. Store in fridge and some in freezer. So it is fresh.
the amount fo carrot juice you will get when eating this butter will be next to nothing. I don't how you feel about soya products (lecithen) but I don't believe all the hype about soy. Like everything else "moderation" is the key. You could buy a egg based lecithen but hard to find.
I don't know about your allergies but have you tried Ghee instead of butter. Alot of people with allergies to dairy can tolerate ghee (clarified butter).
Another thing you could do is just mix the flaxseed oil in a homemade nut butter or tahini. depending on what your body tolerates.