The song is 究極の聖戦 (kyūkyoku no battle in romaji or Ultimate Battle in english). There are two versions of this song:
- The one performed by Akira Kushida (the version Shazam found)
- The one without vocals (the version you are looking for)
According to Anison, the version without vocals is named 究極の聖戦 (カラオケ) (where "カラオケ" means "karaoke").
I searched on YouTube that exact term and I found it (ironically, the video is the same and is uploaded by the same user but it uses the karaoke version of the song):
1. J-Talk translates "聖戦" as "seisen" but many sources use "battle" instead. I have no idea which one is correct.