Broadfoot and Mackenzie were both red-carded for two bookable offences within minutes of each other in stoppage time at Huish Park, leaving Ince short of options for his side's clash with Derby this Saturday.
Ince told the club's official website: "They've let everyone down. They've let the fans down and the club down. My two centre-halves, my stalwarts are now out for Derby.
"Hopefully Cathers (Craig Cathcart) will be back. I might have to throw him straight in."
Blackpool are now fifth in the Championship following their defeat.