Hi again Kendall,
It's probably important to know what else is in the skin cleansing oil. If it has a lot of other ingredients it may not be the best option for putting on a sore. Of the two choices, the lavender may be a better option. Here in Oz, we get the tea tree oil on its own, with nothing added, and it's excellent. Check the label of the cleansing oil to see if it warns not to put it on broken skin, etc.
Re; stinging, that's why I dilute it with water on a cotton ball, to stop the stinging.
And re: Jeri's question, I tend to use natural remedies like aloe, and tea tree oil and honey etc simultaneously, or alternate between one and the other because I'd be very surprised if they interacted in a bad way.