Found a couple of articles about midseason firings and cross checked against Panthers seasons.
While I couldn't find any of those coming directly after losses to the Panthers, a few - Jack Del Rio, Mike Singletary, Bobby Petrino, Steve Mariucci and Dennis Green - came later in the year after losses to us (to be clear, Petrino split with us that season). Back in the 2003 season, Falcons coach Dan Reeves quit right after their second game against us, but the Falcons won that one, sending Reeves off into the sunset with a victory.
Jon Gruden was fired after that hard fought loss to us on Monday Night in 2008, but it wasn't until after the season ended. And as has been mentioned, Raheem Morris was fired at season's end after having been blown out 48-16 by the Panthers in the second to last week of the season. Falcons coach Jim Mora was also fired post-season 2006 after a penultimate loss to the Panthers, but it wasn't a blowout (10-3).
And of course, in the early days, there was Jim Mora. Though Mora wasn't the only coach to embarrass himself after losing to the young Panthers team. Then Cardinals coach Buddy Ryan lost in embarrassing fashion (27-7) after claiming the Panthers stole their defensive ideas from him; a statement which drew extended laughter from Panthers Coach Dom Capers (Capers was a student of Dick LeBeau's zone blitz, not Ryan's 46). At game's end, Ryan was caught on camera leaving the field without shaking the hand of Capers. If memory serves, the game might not even have been fully over yet
Sorry I couldn't find your answer, but hope something in here helps. Links to a couple of helpful articles below:
A History Lesson on the Impact of Midseason NFL Coaching Changes
What Recent History Tells Us About Mid-Season Coaching Changes In The NFL
Midseason Coach Firings are Last Resort for NFL Decision-Makers