Costa will now serve a three-match ban, meaning he will not feature again for struggling Wolves until after the World Cup.
Caretaker boss Steve Davis said: “I haven’t watched it back yet, but obviously the ref went over and decided it’s a red card.
“He’s apologised. When emotions are high it is difficult to discuss things, but we’ll have a conversation with him next week.
“It will open up doors for other players or maybe we will play in a different way.”
Mee had given the Bees the lead with an acrobatic strike from a corner five minutes into the second half.
But Wolves hit back just 110 seconds later when Ruben Neves scored a cracker from 20 yards.
“I think a point was fair, both teams wanted to win the game and it was quite equally balanced,” added Davis.
“I’m disappointed with the way we conceded the goal, we need to do better on those situations. Those are things we need to look at.
“Ruben’s was a great finish and we’ve seen him do that many times. You know where it’s going to go.”
Bees boss Thomas Frank likened Mee’s sensational scissor kick to peak Cristiano Ronaldo.
“He’s probably playing the best football of his life, he looks like he found his second youth,” said Frank.
“He’s a giant in there at the back, he gets a head on everything, and today he scored like Ronaldo back in the day. He deserved three points today.”
A point was at least a boost for Frank following their 4-0 drubbing at Aston Villa last weekend.
“What I asked for was for the bounce back,” he added. “But it was tough to take that they scored straight after we did. They cannot score that. It needs to be impossible.
“Then the bit we lacked was the coolness and quality consistently over the pitch.”
Ivan Toney missed a hat-trick of presentable first-half chances, and Frank revealed his off-colour performance was down to a bout of flu.
“He was sick, on Thursday, on Friday,” he said. “But it just shows his mentality that he still got on to the pitch.”