Barbara is hurt that Claire does not defend her to Allison who is telling lies about her.
Barbara has always been a good friend to Allison and Allison repays her friendship with lies that hit at her very character, which hurts deeply. Barbara is then surprised and hurt that Claire is not moved to defend her in any way, as it seems she is ok with Allison's lies about Barbara as long as she herself doesn't suffer in any way. Claire is a coward and not a friend at all.
Claire is concerned about herself and not about how Barbara feels. Claire thinks it's perfectly reasonable for Barbara to get over the fact that she, Claire, didn't stand up for her and just carry on as before. "Too bad, so sad" is Claire's motto.
That's it in a nutshell.
If Allison is telling lies about Barbara, then of course the mutual friend should stand up and say so.
That being said, none of us here know what is exactly going on. It could be as OP describes, or it could be the opposite, that Allison's accusation of what Barbara did is true, and it is Barbara who is not admitting she's done wrong and is claiming that Allison is the liar. And Claire may not know who is telling the truth.
So, we need more context. OP could tell us what Allison's email said and what the whole thing is about and we could weigh this problem much more accurately. The total lack of any specificity does not allow us to make a judgment for either Barbara or Allison, let alone Claire.