Okay, I checked out the YouTube video by JPL. She looks like an LP-65 Collegiate, but I'm not able to make out an N number under the wing or on her tail. Perhaps JPL has some stills of the LP-65 in their archives in Pasadena...and one will show the registration on the tail or wing. Good luck!
I wasn't aware of a rocket powered Ercoupe, Bradley. That would definitely perform better than a C-85 'coupe. Do you have any photos or footage of this flight? We'd need to see that to identify the chase plane.