Most Go-Pro type cameras rely either on a side mounting to the helmet, or to the top of the helmet or to the chin bar of a full face.
By its design, an open faced helmet lacks the chin bar. The OP wants the camera reasonably close to his mouth, so he can control it via voice instructions. This rules out the side / top / chest mounts. Nobody seems to make a mount to suit.
Necessity must therefore become the mother of invention. Having in mind, the OP’s making of a pie heater, he’s not afraid to experiment, do how about:
Lengthen the two arms, which normally go onto the chin guard, by bonding a suitable plastic ‘strap’ or ‘arm’ to each. Then secure each of the extended ‘straps’ or ‘arms’ to the lower side of the open faced helmet, roughly inline with the lower chin or upper neck. Heavy duty Velcro might well work for this, I guess. The arms of the mount shown might need to be altered, maybe by heating then gently with a hot air gun to reshape them a bit.
Put the open faced helmet on and Velcro the assembly into place.
It might take several versions to get it ‘Just right’, I guess. But the mount shown is not a fortune, Velcro is cheap and suitable plastic for the extension ‘arms’ would not be much.
That said, the main engineering challenges to overcome would be:
A. Not obscuring one’s line of sight.
B. The rigidity of the extension arms.
C. Any jiggling or buffeting, caused by wind blast when riding.
D. The bonding quality of the Velcro.
E. Ensuring that any voice instructions could still be ‘heard’ by the camera.
A fun project for the winter months. Nothing ventured, nothing gained.
PS At a push, I guess some strong elastic or, maybe better still, a rubber strap, might be better than plastic extension arms and Velcro. Put the elastic / rubber band around the helmet. This though might not overcome any jiggling problem. Experiment / trial and error will probably tell all.