Copying Songs From Apple Music to Shokz Open Swim Pro

I recently bought a pair of Shokz Open Swim Pro headphones for swimming. I need to copy songs from Apple Music to the headphones. It accepts MP3, MP4 and several other formats. About 95% of my Apple Music library consists of CD's that I bought, then burned and uploaded into my Apple Music account several years ago. I haven't been able to get any of that music copied over to the Shokz. I've read online that, in order to copy music to the Shokz, it has to be purchased - not just added to my Apple Music account. But I already purchased the music when I bought the CD's. Do I really have to buy that same music again from Apple in order to load it onto the Shokz?