A 2006 study showed that “280,000 people in the U.S. receive a motor vehicle induced traumatic brain injury every year” so you would think that wearing a helmet while driving would be commonplace. Race car drivers wear helmets. But since almost no one wears a helmet while driving a regular car, you probably fear that if you wore one you would look silly, attract the notice of the police for driving while weird, or the attention of another driver who took your safety attire as a challenge. (Car drivers are more likely to hit bicyclists who wear helmets.)
The $30+shipping Crasche hat is designed for people who should wear a helmet but don’t. It looks like a ski cap, but contains concealed lightweight protective material. People who have signed up for cryonics, such as myself, would get an especially high expected benefit from using a driving helmet because we very much want our brains to “survive” even a “fatal” crash. I have been using a Crasche hat for about a week.
Speaking as a SCUBA diver, the equipment is not designed to handle high airflow (such as you need when working hard on a bicycle), so even if the air tank itself wasn't a problem you'd need, at a minimum, a heavily-adjusted second-stage (the one with the mouthpiece) regulator. Possibly a different regulator set altogether. On the other hand, one of the design considerations of a second-stage reg is that the purge valve needs to resist water pressure, including the pressure of swimming; air would generally not have that problem (and you probably wouldn't have any need for a purge anyhow).
Even basic filter masks can cut down on particulate air pollution by a lot. I've spent some time in places with truly horrific air quality - the kind that makes LA seem clear and fresh-smelling - and a lot of people wear something over their face, even just a strip of cloth, when they go out (and sometimes also at night or even all the time). I don't know how practical they'd be at filtering out anything likely to cause headaches in traffic, and they're not terribly comfortable to wear, but it might be an option. Of course, in the US, the most common reason you see everyday people wearing something like that is if they're sick and don't wish to spread germs from their breath / sneezes, so people may be reluctant to shake your hand...