I am not protein sensitive, but coconut oil irritates my hair because it penetrates the hair shaft and traps protein in the hair.
Make sure the oils you buy say cold-pressed, unrefined and/or expeller pressed. Oils processed in these manners contain the highest amount of nutrients and are less likely to contain synthetic byproducts.
Also, if you're protein sensitive, stay away from nut oils (macadamia, almond, peanut etc.). Instead, try oils such as grapeseed oil or castor oil. Try olive oil again, making sure not to use too much.