I know you said you don't do fires during the regular season, just say it was a leadership decision and get it done with. The team didn't even get their full effort today. They have given up. Cut bait, make it known nationally that the job is open, open the checkbook starting at $5M per year, and get to work publicly. I know it makes no difference to keep him or can him right now for a coaching search, but just get it done now so everyone involved can move on and forward.
And if you have someone that has mutually agreed, just have it leaked by an outside source while keeping the name of the head coach anonymous.