About a month before your review session, send this report to your manager. Explain in a cover note that you thought this outline of the past few months' progress would be a handy guide during the review process. Of course, the operator manager is going to helpfully suggest that you spend your time on projects other than writing up things that the manager already knows all about. Don't be dissuaded. Just say that it was no trouble, because all of this material was generated by your direct reports anyway, as a way to allow you to objectively measure their performance; it was no trouble to gather it all together and send along to your own manager.
Formality is your antidote to operators' behavior, and dispassionately dealing with every exchange with them will serve you well. Even melodramatic, undisciplined bullies know when the evidence is against them.
Yes, these formalities will take some of your time. You'll find, though, that this will be a very small price to pay for making obstacles to your team's achievement simply vanish.
And your team members' recognition and rewards will be undeniable.
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