It is strange that the music from Cleopatra isn't available on Apple Music. They have iTunes as a carrier of their music on their website. In your situation, I think I would look to see if the music is available for download on iTunes. If you bought it there, it would merge into Apple Music, so you would still have it available on your AppleTV.
Outside of that, you are looking at buying a new computer or a CD player. You lose a lot of convenience with a CD player, but I don't know how many discs you are talking about. I have SACDs I play on occasion, but most of my music listening is from Apple Music or my server, so I prefer not to deal with physical discs if I don't have to. However, if you are talking about 15 albums or something, I don't think it would be worth buying a computer. If you are talking about 100 or more albums, I would probably look for cheap laptop with an optical disc slot and rip them. If you run iTunes, you can have Apple Music match them so that they are available on your AppleTV when your computer is off, as well.
One thing I would recommend over a DAC would be the Yamaha WCX-50 music streamer. You would need access to a phone or tablet with Android or iOS. However, It has an optical input, so it can work as a DAC (it uses ESS Sabre DACs). It also has an analog input, works with Bluetooth, Airplay (which you could use with iTunes or the AppleTV), DLNA, etc. At this point, I think this type of streamer with a built in DAC offers more options than a stand alone DAC if you don't need numerous inputs.
