You state that you have an HC22E, in which case there is no AV-in and no way of connecting an analogue bullet cam to it. However, if the HC32E does have AV-in (I've not checked so don't know) then a standard analogue bullet cam can be connected to it - but you state that you own the HC22E - are you thinking of upgrading
There are DV bullet cams around, but not that many - and probably not that cheap, but certainly cheaper than upgrading to a new camcorder.
All recording control would be done via the camcorders onboard 'VCR' buttons for either an analogue or DV bullet cam. I'm not sure if the normall 'toggle' button will start & stop recording as well.
I'm not sure about your particular cam, but many Sony cams have a Lanc control input so you can add a Lanc remote control (wired) to control the camcorder without needing to remove it from the safety of the ruck sack etc.
Mark.