The lost data was not recovered, but MAMP was persuaded to cough up its database connections, and Time Machine is now ticking merrily along — although, since it has over a month’s worth of stuff to back up, it’s taking a rather long time. If your Time Machine gets stuck near the end of its backup (e.g. “Backing Up: 3.7 GB of 3.7 GB”) and won’t ever finish, this 5-step process seems to unstick it.
Also, today was our anniversary. Four years without killing each other for the insurance money. Go us!