HIV is extremely low risk, zero for practical purposes; so don't worry about "the big one". (See many other threads on "HIV transmission risk", "oral sex", and "fellatio".) Lip piercing makes no difference.
Chlamydia is rarely if ever transmitted by fellatio; chlamydia of the throat/mouth is rare, and in any case the symptoms came on too soon for chlamydia (usually 5-10 days). Gonorrhea can be acquired by fellatio and symptoms start in a couple days, but the clinic probably would have picked it up; it's easy to diagnose gonorrhea at the initial exam (by looking at discharge with a microscope).
That leave nongonococcal urethritis (NGU) as the best bet, but it generally comes on later (5-10 days, like chlamydia). Another possibility is chemical irritation, e.g. to lubricant/spermicide in the condom, but that's less likely. NGU can be acquired by oral sex. Cases not due to chlamydia, however, are never serious (and may not even mean you have an infection that will harm other sex partners).
Bottom line: close call between gonorrhea and nonchlamydial NGU. Doxycycline is effective against NGU and iffy for gonorrhea. However, the proof is in the response; since your symptoms have cleared up, you can be sure that whatever it is has been taken care of. However, if the clinic prescribed a second medication, you should follow their advice, not mine, and take the drug. In any case, the likelihood of any long term or serious health consequence is about zero at this point, for either you or any future sex partners.
Good luck-- HHH, MD