Update: we crossed the border yesterday morning and submitted the ten-year permit application with my new passport as advised. We received it without any issues, then scraped off the old one (saved it just in case) stuck on the new one, and off we went.
That was a relief! Wasn't looking forward to having to retreat back to Arizona. (Almost had to go back later anyway due to what appeared to be some serious transmission issues with the Sprinter that left us periodically with no power and holding up traffic on those no-shoulder highways, but some quick diagnostic work at the Hermosillo Mercedes dealer sorted that out.)
Thanks for the good advice about the permit.
