You can adjust delay compensation on both Audio recordings or Midi track rendering these days. Usually on a track by track basis. And both on a '+' or '-' basis:
I don't use Ableton, but in Nuendo, there is a slider for adjusting Compensation on each track.
This is also great for Sample libraries where you want to offset start times for long notes, legato times, Orchestral crescendo start times, attack times or short notes etc...
I'm not saying it's the answer to the OP's problem, but it might be worth investigating via the link I left above.