Here's a follow up to my original question.
I used STDcheck.com to arrange for the testing. They offer a 10 panel test for $198.00, covering HIV type 1 & type 2 antibody/antigen (4th gen), herpes type 1 & type 2, hepatitis A, hepatitis B & hepatitis C, chlamydia, gonorrhea and syphilis. After setting up an account and making your online payment, you choose Quest or Labcorp for your blood draw, and then you print a certificate for that lab that describes the tests to be done and to indicate you've already paid for the tests. Because it's a simple blood draw, you don't need to schedule an appointment (which can further protect your identity, if that's important to you); you can walk in, present the certificate, and then wait for your blood draw. I scheduled an appointment just to be sure I wouldn't have to wait. I went to Labcorp, presented the certificate, and they did the blood draw. Total time there was less than 10 minutes. The medical assistant was very nice (and chatty).
I received my results in my account mailbox the next day. (All clear!) That was surprisingly fast but it may be because I live in a major city and lab facilities are abundant.
While the cost felt a bit steep, the experience was easy and quick.