I'm not sure exactly what you're looking for, but I would say the best fat is rendered tallow from pastured beef suet. You can get the suet for free or cheap because not many people want it. The rendering process it actually pretty easy. This pemmican manual has good instructions on how to render the tallow:

http://www.traditionaltx.us/images/PEMMICAN.pdf

It will be a liquid when warmed or solid at room temperature. You might also look into using it to make pemmican. In theory, you can live off pemmican alone, so that might be the ultimate elimination diet.

Caveat: I know nothing about Histamine-Intolerance, so that part of your question might make all this irrelevant.

A good olive oil should be hypoallergenic. (the cheap ones - even extra virgin - are often mixed with soy oil)

It's everything derived from seeds/kernels you should watch out for. Coconut oil and often palm oil have traces of seed storage proteins in it. Those are the guys you really want to avoid. (cross reactivity with gluten)

Olive and avocado oils are made entirely from the fruit, so no storage proteins.

It's not common to be allergic to lard. But again, make sure it's not hydrogenated and from healthy animals. :)

I am not sure if this is hypo-allergenic. However the fat from Wild Salmon is good for weight loss for many people.