I wouldn't do that for so many reasons that I almost don't know where to begin.
First, mics for live use are pretty much exactly the opposite of what you would want for recording. For live purposes, the goal is to avoid feedback and pick up the vocalist with maximum isolation from other sounds. To do this, you capture the sound with a reduced frequency response (heavily rolled off on top and bottom) using a narrow pickup pattern and adding lots of emphasis on certain frequencies to make consonants cut through background noise, etc. For recording, you mostly want something that is accurate. That means maybe a little emphasis in certain frequencies, but not necessarily the same frequencies. Beyond roll-off of the extreme low end to help with handling noise, you generally don't want much roll-off for a vocal mic at all. You generally want a cardioid pattern and no tighter. Otherwise, you get too much proximity effect warmth and everything sounds dull and lifeless.
Second, headset mics are pretty much the opposite of what you would want for recording. The sound quality is good enough for live, but they tend to have a high noise floor, or at least the ones I've heard do. They also tend not to be very flattering to most voices.
Third, single-piece headset/mic combos aren't good for recording because they tend to have way too much leakage, causing bleed of your backing tracks into the vocal recording. Make a change in any of the tracks you've already laid down and you may find yourself redoing parts the vocals as well, which really sucks.
Fourth, USB for audio is a crapshoot at best, craptastic at worst. It tends to be about the buggiest, most unreliable way you can possibly get audio into a computer....
Fifth, wireless for audio outside of live use is a terrible idea in my experience. I've tried using wireless headphones in small rooms and they aren't reliable even ten feet away because of all the problems with signal reflections from all the walls. Ditto for wireless mics. A decent, wireless setup that would work in a small room would cost you a pretty penny (if they even exist in the prosumer space) and really isn't worth it for live use because in a live concert setting, you're usually in a much less enclosed space than you would be at home.
While a good studio mic can be used in live situations with a good engineer, it probably isn't the best idea because it does require a lot of tweaking to avoid feedback problems. Therefore, my advice would be that for your live use, use whatever works most easily for you, but don't even think about bringing any of that gear into the studio (*). Just my $0.02.
(*) Drum mics notwithstanding.
