OP is very useful for the cited reasons. But remember that when we apply oils, lotions, etc. to the skin, this has the ability to "drink' it. And this is why we apply things to our skin (for example,iodide tincture) and derive benefits from it. The mucuous membranes from our oral cavity has the same behavior. Hence, we would not use toothpaste nor other products like sodium bicarbonate. These chemicals would reach our inner system by doing so. The ingestion of sublingual B12 operates under the same principle.
The amount of oil to enter our body would be very minimal under usual living circumstances. But during a period of fast, the body is super sensitive/responsive to external chemicals- beneficial or not.
It is better not to OP during a fast then. Just my opinion :)