How to auto-tune the audio track of a video file to produce a modified video file
Start by looping the video so that you get the timing right - so you end up with that single bark hitting the correct beats for the song.

For the audio, you'll want to then use a DAW or other tool to pitch shift each sample to the desired new pitch. All major DAWs have this functionality built in, or you could use the Pitch Shift effect in Audacity if you like a free option. If you don't know the notes for Happy Birthday, you can easily find them online.

Very quick and easy to do, and you don't need any expensive software or hardware to do it.

And you are right - you are not the first. There are hundreds of these online. Check out this example.