I'll be the first to argue against some levels of H&S nannyisms, but really:
1. they cause a significant discomfort - headaches etc (distraction)
That's a fitting issue. Argue for better fitting.
2. they are ill fitting (distraction)
See above.
3. they can't wear hats or
Which means what, exactly? That they can't keep the sun out of their eyes?
4. appropriate hearing protection when wearing a helmet and
The US Navy cranials allow for soft earplugs to be worn in addition to Mickey Mouse earmuffs. In fact, the last time I was on a carrier deck it was required to do so. If, in fact, your company's chosen headgear doesn't allow for an appropriate level of noise reduction, then your company has chosen the incorrect appliance - argue for a better design.
5. they cause heat stress in hot & humid environments (distraction)
They don't cause heat stress - heat does. The risk/benefit argument comes into play here, and ultimately duty time limits and rest cycles may need to be addressed depending on the work environment.
A private corporation only has to pay out a few compensation claims before the provisioning of head protection becomes an attractive expenditure. If a company has decided to invest in protective headgear and you argue against it in favor of ballcaps, you will lose that argument. If you argue in favor of intelligent design and proper fitting, you may find a compromise that makes more sense.