The Edwards AFB case is in the Blue Book files Case No. 4715.  I don't have a copy but Tom Tulien and others do.  McDonald discussed it in his 1968 House Science & Astronautics UFO Symposium testimony (in the long submitted document of best cases, Case 41, pp. 75, 85).  That's where I first heard of it.  McDonald found out because he was speaking in northern California and met Frank Baker who was the supervisor of Bittick and Gettys (not Cooper whom they knew nothing about).  See McDonald's presentation (the actual date was May 2 not 3).

McDonald's Presentation

I really do not think there were two separate cases, one involving Gordon Cooper.  Cooper's reference to a "camera crew" means they had the camera, and Bittick and Gettys said it was the Askania at Site #4. 

Cooper made up a story years later and claimed HE sent the camera crew out.  The camera crew and their actual boss (Baker) said they never heard of Cooper!

Bittick and Gettys said it got as close as 500 yards which makes it a borderline CE case.  Bittick and Gettys said they radioed their report as it happened in order to try to get other camera crews onto it for triangulation but no one else was set up or manning the other sites -- it was 6:55 AM, and they had gotten there early because their boss Frank Baker had sent them out early (they later, with McDonald, wrongly remembered it was just before 8 AM presumably because that is when tracking operations were normally supposed to begin and other camera crews did not get in place until just before 8 so at 6:55-7:20 there was nobody else to track it).  

The case is in the Blue Book files.  Moreover there is an analysis by a Col. Klein (Kline?) at Edwards AFB in the BB file that destroys the balloon explanation and rejects it based on the known tracking of the balloon.  

Cooper was a nobody in 1957!!!  His name wouldn't have meant beans to anyone then!  He wasn't selected for the Mercury program until 1959 -- two years later.

The case was published a week after it happened, in the LA TIMES of May 9, 1957, (and in an INS wire service dispatch then) long before Cooper came along.  Cooper did not reveal the case or blow the lid on any coverup.  If anything Cooper has almost ruined the case.  

The most likely explanation is a storytelling mentality of exaggerating roles and details, where no one can predict in advance how much alteration the story will get. Cooper probably did see the film and got a garbled story second or third or nth-handed and exaggerated it even more in his mind in the retelling to juice it up.  

But there may be more to it than just embellishment. Cooper's exaggeration of a real case, falsely turning it into a filmed landing with landing gear retracting on takeoff, is very similar to the pattern of Air Force OSI fabrications of other real UFO cases with false landings and aliens using AF officers usually at about the rank of Colonel and designed to discredit the case (Bentwaters 1980 is the classic).  Cooper was an AF officer and I think retired at rank of full Colonel.  Cooper first came out with his phony story about 1978, exactly when AFOSI began a new wave of fabricated UFO stories in an effort to head off what was feared to be another massive UFO flap, this time triggered by the blockbuster CEIII movie.  

Edwards AFB like White Sands was a test range and not a 24/7 air defense base.  Radars were turned on in the mornings for test range activities and turned off at the end of a work day, typically 8-5 operations, a work day, no reason to keep going after 5, everyone's tired and wanting to go home, and no reason to start up in the night either unless a special project required it.  Hence there were probably no radars on at 6:55-7:20 AM.  However if there was a jet interception attempted it was done visually and most likely it was done too late, and the object was gone.  Again it was not an air defense base, no jets were fueled and ready on "strip alert," it would have taken time to get a jet ready and up in the air before the day's operations (had the incident taken place after jets were fueled and ready, in mid-day some time, then a faster jet response could have been undertaken, or a jet might have been in the air already, etc.).  

Cooper's born-again ETH-belief seems to emerge out of nowhere about 1978.