I'll put another nail in the "unfeasible" coffin. I lived in Houston for two years and, being a coastal California native, I never got used to the humidity. It's just miserable to go outside during most of the year, much less have to depend on an outdoor exercise to get to work every day.

It's not impossible, but if you have other options, I would definitely consider a drier climate for a better quality of life considering your situation. The heat isn't really the killer, as biking in deserts like Arizona and Nevada is actually quite pleasant. But that humidity... ugh, I still have nightmares