Why do they NEED Chase behind getting Karlee out?

Because Allen is supposed to be his best friend in the game so it's important that they stick together on all the votes. If you have Allen starting to vote a different way, you could cause a schism or possibly more doubt in his mind.

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They've already had the vote go against how Chase voted lots of times, including when he was supposedly in an alliance and people said they'd vote differently. Every time it happens it makes him more of an outsider and feel less of a bond with the other people who vote against him and he is less trusting of them and how they present themselves. If even his friend in the house Allen never goes along with him then he's really alone there. There's no surer way to make the show feel scripted than to constantly feel like everyone else is doing their own thing and your actions have no effect on the predetermined outcome of the game.

Yeah, Chase would have been totally okay with that, since it was Karlee's fault they didn't have immunity in the first place. Really weird on the part of the producers.