Actually, you won't avoid the copyright issues if you're using someone else's song.
Background info: There are two copyrights involved in a recording: one is the composition itself, i.e., the song and lyrics. The other is the recording, which is a separate copyright.
Strictly speaking, you need a synch license to use someone else's tune on a video, regardless of who created the backing track, and if you were to use the original recording as a backing track, you need two licenses, one for the tune, and one for the master.
The exceptions to needing a copyright are "fair use," face-to-face instruction, and virtual instruction. If you have any chance of avoiding an infringement, you have to argue "fair use." If you're not sure, permission should be requested for the use.
I can't venture a guess as to whether this effort would come under "fair use" since the courts have been inconsistent on the interpretation of the fair use exception, including the USSC. I hosted a studio seminar for the Michigan Bar's Arts, Communication, Entertainment and Sports Law Section a couple of years ago, and after the meeting there was a presentation on the fair use exception, which I stuck around for.
The specialists in the room couldn't agree on the various examples we showed; they could only agree that fair use is an amorphous concept as it stands today. However, I have my doubts as to whether this would be a fair use, since it doesn't seem to meet the criteria. If you really, really want to be safe, write an original tune.
In addition to any actual damages, statutory penalties range up to $150,000 for every work infringed, Infringer pays for all attorneys fees and court costs, and there are criminal penalties. So you want to be very sure you have a fair use.