Everything is dry to the touch now, but I'm encouraging him to continue burning tons of wood and ventilating to the basement for the next several days.
You're exactly right. Don't be afraid to twist his arm, because
dry to the touch is a long way from dry.
If I had a count of all the claims that I've handled that started with, "Mike, I thought I had it dry, but..."
Oh boy.
Keep going. Keep drying it.
Also, the dryout/water mitigation companies generally do free inspections, so he could call them to come probe around with a moisture meter and find out what's all the way dry and what's not.