As you alluded to, I think we can safely assume it was money related, rather than image, as this was a major motion picture at the time produced by Steven Spielberg's Amblin Entertainment (nothing like a brand name to drive up the price).
The request probably never even reached the band. Record labels in the 80s were at their peak of control and we know from their manager at the time, Noel Monk (1978-1985) that they had an "hellaciously bad" contract they had signed as unknowns.
The film makers were trying to create a cultural time capsule of the 80s and in script the tape was labelled simply "VAN HALEN" (p.53). From your screenshot, it certainly looks like "Edward" was written on as a last minute, or who knows, it could have been a pick-up shot after they failed to secure the rights.
Interestingly, I think the reason we got Eddie at all was some quick thinking by the producers. The song we hear is called "Out The Window" from another Universal Picture called The Wild Life (1984) in which Eddie Van Halen scored the original sound track. Eddie described this track as "a bunch of noise".
A look at the credits tells us re-recording mixer Robert Thirlwell (credited as Bob Thirlwell on The Wild Life) worked on both films in the sound department. He was actually nominated for an Academy Award for Best Sound for his work on BTTF.
May best guess is sometime prior to filming he suggested using the tape Universal already owned as a way of keeping the spirit of the original script, while working around the rights issues.
Bonus trivia: Lea Thompson (Lorraine Baines in BTTF) also starred in The Wild Life, along with Eric Stoltz...who was the original Marty McFly! She was scouted on set while the producers were watching Stoltz.