There's just no room for him in any top six or bottom six role. It's the problem they had with Moulson and Purcell. While they've broken out to become top six forwards, I don't think those players, including Bud, would be in such a role with a deep, playoff contending team.
The Kings have let far more talented players than Bud go, Moulson and Purcell are a couple examples, they dealt away Cammalleri who was one of the best goal scorers on the Kings for a period of time, but if he's not scoring, he's not helping his team out much. The same can be said about Jason Blake who also broke out with the Islanders but again, he isn't going to help out much if he's not potting goals.
If players like that struggle in the playoffs, your team is going to be stuck in a hole. Take Tomas Sandstrom as an example. He was one of my favorite Kings and was one of the best goal scorers on the team during the Gretzky era, but as we saw in the playoffs, if he wasn't scoring, he wasn't doing much else to help the team win. And he was pretty bad during the Finals (and I blame him for being out of position and not covering Desjardins, twice, in OT in Game 2).
It's time to move on from Bud Holloway. Thankfully this isn't the 2003-04 or 2006-07 Kings that were depleted of any depth and were desperate to add bodies to fill anything that resembled an NHL lineup. They're far past that and losing out on him isn't going to hurt this team in the least.