I looked into this a couple of years back and don't think any Epsons are capable (not absolutely sure on this) but most of the Canon Pixma PRO printers are. I know the Pixma Pro-100 is because I have one. It's a relatively low priced printer which is advertised in specials from time to time where it can be had for under $100US (not seen any of those specials lately so keep your eyes open).
Canons use a small tray which the CD goes in and it is placed in the back of the machine where it is moved under the print area. Of course,
printable CDs, DVDs, or BDs are needed (which you mentioned you have)
I have a stock of printable labels (NEATO seem to work the best) which I still use for quick labeling jobs and as TonyW mentioned, they work well. I use a DIY Photoshop template for those, which lines up the artwork for jobs, The downside of labels is that you can't print to the entire CD and a 1.4" hole exists at the center, letting the hub show. A lot of printable CDs allow printing almost to the CD hole and result in a more professional looking job but labels do look good if you discount that issue (a bit dicey to attach but you get good at it with a little practice - don't use any jigs that are sold for the purpose - just line up the label using the CD hub as a reference, watching the edge position at the same time).
https://www.usa.canon.com/internet/portal/us/home/support/details/profession...00?tab=manuals