I would insist on antibiotics. If it is a tick bite it could be Lyme Disease. My son was bitten by a tick and was very poorly indeed! He was sure he had Lyme Disease and insisted on antibiotics, ( wasn't given them! ) but was given a blood test . The blood test was sent to a tropical disease centre and sure enough he had Lyme Disease! When the test results came back his GP gave him antibiotics - 3 months supply!!?.He was very lucky indeed not to have life changing conditions, was so ill, if antibiotics had been given sooner my son would not have been so ill!
A tick bite does not always have the bulls eye marking. Don't wish to scare you and of course this may not be a tick bite, but forewarned is forearmed!!!