When we ate there, a woman has a shellfish allergy and something else that i did not hear clearly, but think was mushrooms. My view was that they were very careful.
Before the orders were taken, a server came around and asked each person individually if they had any allergies or food related issues that the chef should be aware of. Then they took the orders, asking again about allergies. Then, when the chef came out, he asked a third time and spent some time talking quietly with the woman who had allergies.
When he started cooking, he cooked the food first for the person with the allergies, so hers was cooked on a clean grill (more about cleaning later). Another person and I had shrimp, which he cooked on an area of the grill that was not very close to the person with allergies.
Each person got their own little bowls of condiments, so no chance that someone else would dip their items into that lady's condiments.
They did have squeeze bottles of some kind of oil that the chef used for cooking with. I don't know what kind of oil it was, possibly peanut, but maybe not since the grill is extremely hot and peanut oil does not hold up to heat well. They also used a squeeze bottle of something else - may have been soy sauce. I remember the chef did not use that bottle on the lady's food, so soy may have been an issue for her.
I highly doubt they would use butter, since that burns easily (and I don't think dairy products are used much in Japan).
At any rate, they should be able to accommodate your needs since they cook the allergy items first.
After cooking, they did a kind of cleaning ritual on the grill. It was scraped, rubbed, washed and rinsed repeatedly.
Pork may have been a different situation. Some people do not eat pork for religious reasons and would not be able to eat food cooked with or after pork. Unless an allergy was very severe, I think they would just cook the food first for the person with allergies as they did while I was there.
