I have some family experience with Lyme and meningitis. I'm confused.
A blood test is not useful for a recent tick bite. It takes 4-6 weeks for the antibodies to be present in detectable quantities. For Lyme, you observe the possible site of infection for a few days, if it is swelling or reddening (with or without the characteristic "bulls-eye" pattern), then you get antibiotics. Doxycycline is popular, but it's a long (21-day) course, so a good doc will not prescribe it casually.
Meningitis is either bacterial or viral. The true test is analysis of spinal fluid (this one goes to 11!), but it's a very fast-moving and dangerous infection. Regardless, a blood test is unlikely to be conclusive.
You can save the tick and send it off to a local lab for analysis. That'll tell you with a high degree of certainty if the tick had the spirochette. In some parts of the country, most do. It won't tell you if it has been transmitted to you, of course.
Disease transmission typically happens after the tick has been attached for 36-72 hrs.
Those of you who haven't lived in deer tick country might not realize how tiny the things are this time of year. Pinhead sized or smaller. Easy to miss even if you're diligent. No solution for it except to be more diligent, however.
Good luck with the doxycycline. I don't understand why your PCP ordered a blood test, but the meds make sense.