I was thinking helmet mic for voice recognition. It wouldn't really need a word as such, just a recognisable sound. Better than having to find a spare hand/finger and hit a button. That said you could probably find somewhere to fit an easy hit switch of some sort that would do the job.
It may be better to think in terms of a helmet mounted camera than a fixed car mount. At least that way you could have some control over the composition. However I think that would probably be quite a challenge on the approval front unless it was a small video style 'bullet' cam device.
Have you considered buying a Morgan 3 wheeler and carrying a passenger to take the shots?