Hi linkb - welcome to our community!
The bullseye rash is a lyme disease symptom. Unfortunately 2 weeks isn't enough to treat it.
Yes, continue to pursue it...you've caught it early and your chances of quicker remission are greater if you treat it now.
According to the ILADS guidelines - you should be treating until symptom/sign free for at least a month.
with a 6 week minimum
If you can't get more antibiotics, then try an find a LLMD and explain your situation...so maybe you can get in right away...or be put on a cancellation list.
www.lymenet.org/BurrGuide200810.pdf