Complete re-edit to avoid speculation:
An employee lied to a new customer completely behind the back of the PM/leads/management that he/she was not being allocated (he/she actually was 100% allocated) and therefore available for a new project which he/she liked better. Nobody knew about this conversations, the new customer expects delivery and the old team is stretched thin by this (and the old customer may have some projects which run late, potentially causing financial damage). There are circumstances which prevent management level from communicating clearly, but I was one of colleagues in the team to whom she bragged about how great this move was, and how cool it was to lie also to PM and management.
Would it be appropriate to have a clarifying conversation with the technical side at the old/new customer (both in one company), to clearly state that they should be very careful about promises of this employee in the future, pointing out that he/she intentionally lied, even if this information may damage the employees further career in the projects at the customer? Up to now our team/management was playing nice and took the blame under "there was some miscommunication".