Cooper said: "I have to be careful what I say, but if that's a sending off then we're going to be ending up five-a-side.
"It was never a sending off, it was a yellow card maximum. He never left the floor, he's made an attempt to play the ball and trod on the boy's foot - it wasn't high.
"It's a game-changing decision for us. I've had to bring players off that were playing well, who are disappointed.
"We've had fans who have travelled and paid good money and we get decisions like that - it's totally wrong.
"I think the referee will probably look at that and think he was a little bit hasty, he didn't give himself any time to think about it, he just pulled the red card out straight away.
"I think we needed the second goal, but I was confident that if we'd had 11 men on the pitch we'd have won the game."
Swindon carried the greater threat in the first half and Pritchard put them ahead after 17 minutes, curling a brilliant 25-yard free-kick into the top corner for his first league goal.
They were reduced to 10 men six minutes after the break when Byrne appeared to stamp on Williams' foot as the Dons defender cleared the ball.
Bamford equalised for the Dons just four minutes later, racing on to Darren Potter's defence-splitting pass to poke the ball past Wes Foderingham.
The Dons pressed for a winner in the closing stages but substitute George Baldock had an 89th-minute goal controversially ruled out for offside.
MK Dons manager Karl Robinson agreed with his opposite number that it was a harsh decision to send off Byrne :"I generally look at my players and there was no reaction from them to say it was a bad tackle.
"He has gone in to challenge for the ball, but it's a lazy challenge. You can get seriously hurt from it but, for me, I can't see that being a red card.
"I've had people sent off for an awful lot less, but if I was in Coops's shoes right now I think I'd be disappointed."