Any doctor can do an oral swab for gonorrhea and chlamydia. Syphilis testing is a blood test no matter where the possible exposure is.
Most STD clinics don't do oral testing because they operate on shoestring budgets and can't afford the extra supplies. If you had a known oral STD exposure, they might just treat you instead. If your doctor won't do it, it's because they don't know how.
That said, most experts don't feel that one oral exposure doesn't warrant testing. It's much lower risk than vaginal or anal penetration.
What's making you want to test?