I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear that this happened.
When the transfer is complete, it should be back-dated to 6/1, which means that your interest will be adjusted. If that doesn't happen automatically, yes, you can require them to do a manual adjustment. Hopefully, that can be done without filing a lawsuit - any manager at the bank should be able to take care of it for you. One way to avoid suing is to send a letter to their legal department explaining what happened and giving them a firm deadline for fixing the problem.
But if that doesn't work, or if you'd prefer to go to court, yes, you can sue them for the missing interest (and the missing MONEY, if it doesn't show up before court).
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