Can you use the Diplomat feat through your Familiar with the Voice of the chain Master?

So the Feat goes:

If you spend 1 minute talking to someone who can understand what you say, you can make a Charisma (Persuasion) check contested by the creature’s Wisdom (Insight) check. If you or your companions are fighting the creature, your check automatically fails. If your check succeeds, the target is charmed by you as long as it remains within 60 feet of you and for 1 minute thereafter.

Can a player trigger this effect through a familiar when he speaks through it? I don’t want to deny my player this "because I said so," but it totally breaks any notion of balance for him to (for instance) send his familiar to god knows where, find someone important, pretend to be a great wizard trapped in a pseudodragons body, charm them through diplomat, then make them do god knows what.