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").
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Do you mean after Bakugo reassures Kirishima that "if the enemies are small fries like these, then they will be alright"? The song played after that scene is called "Hero A". I hope this is the one you are looking for. 😉