The temperamental Cameroon defender conceded a penalty following a foul on Clarets substitute Steve Thompson and was booked for his indiscretion.
Graham Alexander, 37, rifled home his 11th goal of the season - his 10th spot-kick - and in the closing stages Bikey became involved in a tussle with Robbie Blake which saw him shown a straight red card for stamping on the Burnley striker.
Yet Bikey refused to go quietly and removed his shirt before throwing it to the ground in disgust in a remarkable show of petulance.
"The red card was fully deserved - there is no defence whatsoever," said Coppell.
"We just don't accept that at our club.
"It was a hugely emotional reaction given the five minutes he had had.
"You have got to remember he was fouled first (by Blake) but there is still no excuse for his behaviour."
Bikey will now miss Tuesday's decisive second leg at the Madejski Stadium but Coppell was also quick to point the finger at referee Martin Atkinson.
Coppell was not convinced that Burnley's penalty should have been awarded.
"I've seen it five or six times on television and it's still unclear whether it's inside or outside the box," said the former Crystal Palace manager.
"I think initial contact was certainly outside the box and the fall was obviously exaggerated, as I think any forward would do to be honest.
"The referee is 10 yards away and there is no way that he can say categorically that it's inside the box and a penalty.
"There was no input whatsoever from the linesman and I have to say it was a 'guestimate' at best."
Coppell could also be without Kevin Doyle after he sustained an Achilles injury and was replaced in the first half.
Burnley manager Owen Coyle also has injury concerns after Wade Elliott missed the game and Joey Gudjonsson and Chris Eagles were substituted having picked up knocks.
Coyle acknowledged his side failed to play to their potential in what was their 59th game of a marathon campaign.
Coyle's men have enjoyed a thrilling campaign which has seen them dump Fulham, Chelsea and Arsenal out of the Carling Cup and come agonisingly close to overcoming Tottenham in an epic semi-final encounter.
Yet they struggled to make any impression for the majority of this game until Alexander's late intervention and Tuesday's decisive second leg remains finely poised.
Reading should have forced home their first-half dominance as Clarets goalkeeper Brian Jensen denied Jay Tabb, Marek Matejovsky and Bikey while Martin Paterson hacked Kevin Doyle's header off the line.
Coyle said: "There's still a lot to play for, there's no doubt about that.
"The game wasn't really what I had anticipated in terms of two good football teams getting the ball down and playing.
"I thought Reading came with a gameplan to stifle our threat and all credit to them.
"They were relatively successful in doing it, but Brian has also made an outstanding save and Martin Paterson has kicked one off the line.
"But I can't ask any more of the players and the courage, the desire and the hunger they have to win football game is so evident.
"We know we have to play better than that and that's what we'll look to do."