The reason Eddie is getting worldie performance out of what is still essentially Bruce's team is because he's got them playing as a team. Taking responsibility, and taking your licks is part of that.
So no, he's got to drop him. Hopefully not for long, but Eddie has to make the point or risk undermining the unity that's doing us so much good.
Otherwise - let's say next week someone else is late to training. If Eddie drops him, then he's got a fair point that Joe did far worse and got away with it. If Eddie doesn't, then discipline is shot.
This is a serious issue with serious consequences. Most jobs I've had I'd have lost if I'd been caught driving over the limit. So a week's wages and a symbolic match out of the team is hardly over reacting.