So, a few things:
1. Based on your description, I am a bit concerned that there is a secondary infection. Therefore, this makes this situation now well worth a vet visit to explore the area, ensure the tick is fully removed (no retained head), and possibly to start some oral antibiotics.
2. You can apply some neosporin to the area a few times a day in the meantime.
3. Yes, there is always a chance for transmission of a tick-borne disease like lyme, ehrlichia, or anaplasmosis. However, there is a simple blood test (some vets can even do it right in their hospital) to test for these diseases. If diagnosed early, treatment can be very effective with no complications occurring.
I wish we could easily treat at home, but with the need for possible antibiotics, the possible need for a tick head to be removed, and for the consideration to do a simple tick disease test, the vet visit is important.
I hope that information is helpful.