Because codes are arbitrary, and vary from administrative area to area, I'd suggest working by pilot.
Caveat: I'm talking about building codes here. In many places, FAR (floor area ratio) / density / zero lot line zoning codes may be a larger obstacle to generating good, cheap housing.
Design a project that attempts to provide low-cost, safe, bare minimum housing. Then start the approval process. Empower a panel to take notes on what happens and make recommendations on waivers for each issue. And optimize for cost -- including approval time!
Then bring those back to your (the mayor's) desk, and if needed, back to the local populace.
It seems a fair bargain to me: in exchange for quality, cheaper housing (a social good), government agrees to relax less important codes which drive up costs (allowing the developer / builder to make a profit on such housing).
[Soapbox] Part of the issue with deregulation is that it's often done in a top-down, political way (by necessity).
When in reality, the deregulation that matters is infinitely nuanced (e.g. "The local water board wrote a requirement for not more than 20° bends in drainage systems, but by upsizing pipes or using catch basins, the same flow could run up to 70° bends.")
And the only way these things are recognized is through actual projects (e.g. Mr. Steve's).
But I actually like your answer overall.
Rough week.
That's a rather broad suggestion and I feel like it gives me nothing to latch onto. Pretend I'm your mayor. What, specifically, do I need to keep and what, specifically, can I relax?