The director, as the add on is called, would have to be in correct position to interface with the existing antenna. That said, if I was in a tight situation where I felt I had to have some improvement, I might try the add on to your existing antenna. Attach it in the same plane as the antenna elements, on the side of the antenna that faces the source. That could be done temporarily to see if it worked, and it might, then you could devise a mounting arrangement for more permanent use. You might move the "add on" closer or further from the existing antenna until you got it tuned in.
If it does not work, then upgrade to the Winegard Batwing, and install the booster on that. I think we have established that will work.
Might be a fun project. Report back if you try it.
If I am all wet on this, I hope someone more savvy will step up and keep you out of trouble.

I am not familiar with your antenna, so have no clue if mounts are compatible.