Blutooth audio choppy during backup
This is one of those trial and error situations that you'll just have to explore.  I know for sure that Read/Write from backing up do consume both CPU and Memory. So if your backing up a ton of data and especially if your Hard Drive is the mechanical kind not SSD then in my experience playback of music from iTunes or files actually on the computer.
The only way to really know is first you must take out some variables. If your internet connection is solid use youtube or another streaming service so that we can somewhat negate the read/write of a file being played off the Hard Drive (buffering does do some of that for streaming but I'd wager not as much). Ok now to test!
1. Make sure your Mac has all it's updates and firmware updated. With very few exceptions all newer bluetooth revisions can speak with older bluetooth. Backwards compatible that is. But it would be good to make sure it's not an OS issue first.
2. If you have another bluetooth speaker try and recreate the issue by using it instead and then doing a backup.  If nothing happens then great you have just figured out that it's probably your bluetooth speaker that's failing somehow. (which would be weird but hey stranger things have happened) If it still cuts out move to step #3.
3. To figure out if interference is the cause get the longest cord you have that connects to your external backup and move it as far away from the Mac. Be sure to put it away from &quot;line of sight&quot; path of your bluetooth speaker.  Now do a backup. If it was the issue I'd really consider getting a new backup drive as that would be crazy that it was putting out that much interference.
I hope this gave you some guidance on maybe where to start. I know there wasn't a solution nor was there a bunch of technical OS stuff you could try because honestly there shouldn't be a correlation to bluetooth acting up when doing a backup between the OS and your speaker. I listen from a bluetooth speaker all day with 4 external backup drives and a ton of audio gear (grounded mind you but still making tons of interference).  So the only thing I can see it being is either speaker, OS or Hard Drive hurting while doing a backup so it can't play properly.