Distilling it is no problem, getting it HBr free is a challenge. Everything has to be bone dry, however not even that's enough. Below is a trick I used for ester enolate Claisen rearrangements.
Distill TMSCl fresh. Take a known aliquot, then carefully add to a known volume of Et3N. Centrifuge down the Et3NHCl precipitate. Supernatant contains acid free TMSCl.