The West Yorkshire-based official, making his first visit to Dean Court, had a day to forget as Cherries were held to a 2-2 draw by League One rivals Charlton Athletic.
Haywood infuriated Cherries supporters when he allowed Scott Wagstaff's opening goal to stand after the winger had appeared to handle the ball over the line.
But Fletcher and Bradbury were upset when Haywood failed to spot an infringement in the lead-up to Bradley Wright-Phillips's equaliser nine minutes from time.
The Cherries management team, together with more than 6,000 home fans, were convinced Christian Dailly had fouled Jayden Stockley just seconds before Wright-Phillips pulled the trigger.
Following the final whistle, Haywood left the pitch to a chorus of boos and was flanked by a posse of stewards.
Fletcher told the Daily Echo: I have watched the DVD and Dailly committed a blatant foul on Jayden. Jayden cushioned the ball on his chest and he came through him with two hands and a knee in his back. He munched him from behind.
I can only think he didn't give it because, about 10 seconds earlier, they put over a cross which hit Rhoys Wiggins on the arm. Charlton appealed for a penalty but he didn't give it. It wasn't a penalty anyway.
Asked for his view on Wagstaff's disputed opener, Fletcher said: The ref said he thought it might have hit his hand but that it was more ball to hand rather than the other way. I think their player admitted it came off his hand.
He ran in at pace and it rebounded on to him. You hope they go for you and the Charlton player will probably say it was unintentional. You just have to deal with these things.
Boss Bradbury said: I can't comment on the first one because I had someone in the way. For the second one, Jayden did what he should have done. Dailly had two arms in his back and pushed him to the ground. Jayden is a strong lad and it takes a lot to push him over. We were very disappointed with it.
Despite the controversy surrounding both Charlton goals, Fletcher acknowledged Cherries should have taken more than the two chances converted by Marc Pugh and Danny Hollands, particularly during a frantic finale when Bradbury's men laid siege to the visitors' goal.
Fletcher added: In 21 years, I have never seen an end to a game like it. I would imagine people who have been coming here for 30 or 40 years would say the same. It was real edge-of-the-seat drama.
Our supporters were trying to suck in the ball for us and I was just waiting for the net to ripple. I thought we had scored on seven or eight occasions.
It was one of our best performances for a long time. I'm not taking anything away from Charlton, they defended resolutely. But to say they rode their luck would be the biggest understatement of the season.