To mute a specific sound caused by something not UI-related (like noisy battle pet idle sounds), you need to substitute the sound in the Data/Sound folder.
The tricky part is identifying the sound file and knowing where to put it.
wowhead has a sound database that can sometimes get the filename of the sound, but it doesn’t describe what folder the sound should go to and it’s not complete.
Finding where it goes to is often just brute force attempting different folders, finding an addon that replaces a similar-genre sound and using that folder (like if you find one for muting another pet it’s likely the same one),
Once you’ve identified the sound and where it should go, like SPELL_WR_WildStrike_Cast into Data\Sound\Spells, you can create a 0-length .ogg file of the same name in the appropriate Data\Sound folder (creating the sub-folder if it doesn’t exist).
You don’t need any special software to create the mute sound. On windows if you navigate to World of Warcraft\Data, then create the Sound folder if it doesn’t exist, then the Spells folder beneath that if it doesn’t exist, and right-click a blank area inside, create a new text document and name it SPELL_WR_WildStrike_Cast.ogg, then leave it at that, it will work to mute the sound.
World of Warcraft\Data\Sound\Spells\SPELL_WR_WildStrike_Cast.ogg