How to rotate player character while shift lock is enabled?

The gun system in my game forces shift-lock to be on. The problem I am having is with the animations pointing not exactly to the center. A video about the issue:

I tried changing the root part’s CFrame, but while shift-lock is on, it appears that minor changes are overridden. Only when I change the angle significantly (like by 180 degrees) does it take effect. So, I need a solution that either somehow goes around this, or rotates the shift-locked camera a little externally.

Thank you.