Using Content Aware Fill after Warp Stabilization

Content-aware fill does not recognize effects applied to the footage. That is what is throwing off your workflow.

You probably want to work the other way around. Content-aware fill first, then pre-compose, then warp stabilize your nested comp. Warp Stabilizer bends things to fix motion and it is almost always better to do compositing before you stabilize. Then you can stabilize the entire composite as a comp (pre-comp) and get the same warping on the entire frame. I don't use Warp Stabilize very often but when I use it on a shot that needs other effects or other layers added to the comp I do all the compositing, match moving, and effects on the original footage, then pre-compose and warp stabilize the final shot. 

If that doesn't work for you, then try pre-composing your stabilized footage or pre-rendering it before you try content-aware fill. No matter what workflow you use, you'll have to pre-compose or render the composite before you do the second step.