You might want to consider getting an Amateur Radio license. There are a lot of nifty things in the radio world that Amateur Radio would open the doors to.
That having been said, below is a link to a cheap(!!) headset that I have used. I think I have 3 of them around here. One of them I took apart and bent the headband to fit my bike helmet. They are a "behind the head" style headband. Though that works better than a conventional "over the head" headband would with a helmet, I can't wear the behind-the-head style with the helmet. The helmet still pushes it down. So I made the band fit the helmet instead.
Behind-the-head style headset for UV-5R UV-5RA UV-5RA-UU UV-5RA-plus 409shop,walkie-talkie,Handheld Transceiver- Radio
Both the mic and speaker/earphone work surprisingly well for such a cheap headset. The speaker can get way louder than what I can stand. In fact, I should put a resistor in-line with it to knock the volume range down even further because the main handheld radios I use it with (Anytone d868's - DMR/Analog VHF/UHF radios) have segmented volume settings (they don't have pot control - they have increments 1-10 or 1-20 or so) and the lowest level is still too loud through the headset.
The vendor is an over-seas (china) vendor - ordering from them can take 2-3 weeks to be shipped and your orders are usually labeled as something goofy to get through customs (especially anything with lithium batteries).
As far as radios go - there are a lot of options. I am not all up to speed on FRS radios (unlicensed), however there may be a way you can tie one of these headsets in with one of those. The headsets I have (along with some other speaker mic type accessories) use a round multipin connector and then an adapter from that to the style plug any particular radio uses (most of my handhelds use the K style plug as the headset in the link takes you to). If an FRS radio has a plug on it that you can get an adapter (or fixed connector) to work with on the headset then it is conceivable that would work just as well. You just have a lot more flexibility on the Amatuer/Ham side.
Another note is an Amatuer or FRS radio will require a PTT button. Yes, lots of radios have VOX (voice operated transmit), however when riding VOX will drive you crazy. It will trip when you don't want it to. So my 2 cents worth is to throw VOX in the can and run a remote switch or button. To that point - you can get a velcro strap mounted PTT button for the above headset/adapter system. I mount that to my handlebar to where I can press the button with my thumb mostly. That way I am not needing to do anything more than hold the handlebars, maybe just in a slightly different position, but that is where the velcro is nice - it can be moved around to change positions if need-be.
I hope this helps give you some ideas.