I think this hearkens back to the discussion about what skill one should use to determine how well a character can cook, and is ultimately going to pan out the same.
Performers, grifters, and regular folks are going to approach accents and mimicry from different avenues, with a variety of strengths and weaknesses. A comedian who does drop-dead impressions of well-known people may have no capacity to replicate an accent unless they studied a speaker of that accent, and then they'd be mimicking that speaker. As with the grifter from "Leverage," someone may have a fantastic aptitude for improvisation and deceit, but be unable to really act to a script.
The solution is Dresden should be to figure out what that character *is* good at (lying? improvisation? mimicry? bluster?), buy an appropriate apex skill, and then get Stunts which either:
a) move one or more Trappings to that apex skill to simulate mimicry/etc.
b) provide a bonus when mimicking/etc.
c) allow that Apex skill to Complement skill checks having to do with mimicry/etc.
So if you had an actor as a PC, give them a decent (+2 to +4) Performance, a Stunt to give +2 specifically when using Performance for acting, and maybe other Stunts moving Trappings from Deceit and Rapport over to Performance. There's just so many ways to build it, and different tables may prioritize different skills for these things.