No. Cavallo "misremembers".
In fact, the recording history of "Kiss" is well-documented and has been told numerous times. One of these stories can be read on the MusicThing blog, and of particular interest is that page's November 2013 update at the bottom by Prince historian Duane Tudahl.
A song for Mazarati
Tudahl explains that "Kiss" started out as a track for Mazarati, a band on Prince's Paisley Park Records:
On Sunday April 28, 1985, Prince was recording the tracks “In All My Dreams” and “Evolsidog” at Sunset Sound in Studio 3, and David Z. and BrownMark were working on songs for Mazarati in Studio 2. “We were in the studios next to each other, checking each other’s progress,’ recalls engineer Susan Rogers. “And at some point they said that they needed a song.
Prince stopped what he was doing, I remember this very clearly, he had a little boom box, a little pale green, Sharp one which he had to record ideas on, and he took it and an acoustic guitar to the next room, put it down, put in a blank cassette and he pressed record. On the acoustic guitar he then played "Kiss". It took a few minutes to get the lyrics; he recorded the guitar on one track and the vocals on the other track. He then took out the cassette and said: 'Here, finish this off.'"
This recording is actually circulating among fans and has been available on bootlegs for decades.
The recording process
Note that a significant part of the creation of the unique sound of "Kiss" didn't even involve Prince:
The song was transferred to 24 track and engineer Coke Johnson went to work. “I took that to studio 2’, recalled Johnson. ‘We started fiddling around with it. We used the same changes, but instead of using that acoustic guitar, we ended up gating that guitar and the hi-hat. That is the weird sound you’re hearing. It’s playing the same rhythm the hi-hat’s doing, but it’s doing the changes the acoustic guitar did. That is one of the biggest hooks with it. David thought of the idea, and I hooked up the gate. He was flipping the switch to throw the delay in and out, and actually created that sound for ‘Kiss’.”
Then the band got involved and work on the song continued:
Musically, the piano part was lifted from Bo Diddley’s 'Say Man,' and the backup vocals from Brenda Lee's 'Sweet Nothings.' Terry Casey’s vocals were added and most of the band left after 11pm. David Z., BrownMark, Johnson and Tony Christian stayed until the following morning adding depth to the song.
The following morning David Z. wasn’t happy with the results. “We were trying to build a song out of nothing, piece by piece. It was just a collection of ideas built around the idea of a song that wasn't finished yet. We didn't know where it was going. We were getting a little frustrated, and we were exhausted.”
(Mazarati's version is also circulating among fans.)
Prince takes it back
And then Prince got to hear it:
Eventually, the results were revealed. Coke recalled the reaction. “We played it for Prince, who went ballistic, went out to the basketball court playing it loud on the ghetto blaster. He pretty much said: ‘This is too good for Mazarati.’ It pissed us off as we had been up all night working on it.”
He then took the song back:
Prince took the tape back into Studio 3 and began making his own changes. He quickly eliminated BrownMark’s bass [...] and added the James Brown/Papa’s Got A Brand New Bag style electric guitar riff as well as his vocals, which he recorded an octave higher than Terry’s. [...] The track was completed on Monday April 29, 1985.
This article @ MixOnline explains the origins of the track from David Z's perspective (and is apparently the source of several of the quotes in Duane Tudahl's story as quoted above).
I also recommend this article @ Sound On Sound, which features more background on David Z.
Inclusion on the soundtrack
Of note is that "Kiss" wasn't even originally included on Parade (Under The Cherry Moon's soundtrack) by Prince:
Prince began working on the album in mid-April 1985, shortly before the release of the previous album, Around The World In A Day, and began by recording the drum track in sequence of "Wendy's Parade" (late renamed "Christopher Tracy's Parade"), "New Position", "I Wonder U" and "Under The Cherry Moon", all in one take. He then layered other instruments over all four tracks.
A first configuration of the album was compiled on 1 May 1985 (detailed below), but several changes, including the removal of three tracks and the recording of six additional tracks, took place between then and the final release.
This is that configuration, which doesn't include "Kiss":
- Wendy's Parade
- New Position
- I Wonder U
- Under The Cherry Moon
- Others Here With Us
- Life Can Be So Nice
- Sometimes It Snows In April
- Old Friends 4 Sale
- All My Dreams