Keith was upset by the cold way that Carter kicked him to the curb after he had served his purpose for him.
You've seen Guy Standeven in something because the man was in everything.

In addition to everyone else's great points, it's probable that Keith did not realize exactly with whom he was getting involved or the potential deadly consequences.
If Keith knew that hanging around with his friend's brother and asking a few questions to patrons at his bar was going to get him broken bones and potentially killed, one doubts he would have done it.
In short, he's also mad bc Caine didn't tell him what exactly what he was getting into.

Keith also says something about his girl coming home from Liverpool. Maybe she was away at the art college and was coming home on vacation to find him beat up and "non-functional" which he feared would cause her to dump him.

You remember the line when Carter throws him some money and says something like "buy yourself some Karate lessons?" It is Carter's attitude that everyone can be bought and paid off with a few notes. This attitude is what makes Keith so sore. In other words he is insignificant and disposable.

I think it was more the breezy way Carter was able to brush off the entire situation. It really looked like Keith got his ass kicked, and Carter's attempt to "pension him off" was pretty cold. Like everything he does I guess.
But you're right it's fun to play good guys and bad guys with Carter, but Keith clearly didn't appreciate the possible consequences.