I have a soft spot for Lavender Oil. Not Essence of Lavender Oil, but the real thing. Before I say anything further, I am not a doctor or qualified in any way to give the following information. If you choose to follow this advice, it is at your own risk. With that said, everything I am about to say I have tried on myself and believe it does work.
Lavender Oil is a multi-use item. I've put it on as perfume, and headaches have lessened in severity. I've put it on mosquito bites to relieve itchiness and redness, and it has worked. I've put it on burns, and it not only helped the pain but the burn itself healed faster with no scar.
Like I said earlier, you want actual Lavender Oil. Essence of Lavender Oil has at least one ingredient that will not help a burn. In fact, it will make the burn worse. I would assume that it would do the same for a bug bite.