Should I get a second opinion when my son has a bullseye rash from a tick bite and she does not administer an antibiotic right away?

My 3 yr old son just developed a bullseye rash on his arm yesterday. I first noticed it 2 days ago when it was just a raised red spot..looked like a very large mosquito bite. Then yesterday it was much larger and the exact picture of a tick bite bullseye rash. I took him to the Dr immediately and she said it could be a tick bite but without a tick present she was unsure. She did bloodwork to check for Lyme disease but I think she should have at least started him on an antibiotic to play it safe. I suggested that and she refused. I am very worried…should I take him to get a 2nd opinion? If the BW can give a false negative…doesnt being cautious with an antibiotic make sense?!