These are the ones you are most at risk for when allowing non-penetrative skin contact. Which, by the way, you can get several of these just by being in the vicinity, so seriously dude, stop the near miss grinding as well. Unfortunately, there is no test for HPV for men. You're just going to have to wait that one out and see if you get warts, or you give your wife cervical cancer.
Why am I being hard on you? Because you were playing with fire instead of using your big head. If a provider is engaging in risky behavior, leave. Do not fight over the money, just go. If she's done it with you, she's done it with countless others.
Here's how else you can protect yourself: if you're under 45 years old, you can walk into your local pharmacy and ask for the HPV vaccine-it's available to men too! This will minimize your risks for HPV, which can still be caught with condom use since the wart versions live on the skin (genitals, thighs, buttocks, ass, etc.)
Also in the future, you may want to look for providers that offer FC2 condoms. There's not many of us, but I very much believe the extra coverage around the outside helps minimize skin contact. (Plus they feel better for most men!)