Definitely the amp in my opinion. A warm amp will make all your sources warm, if you ever add a tuner, phono, tape, etc. Getting a warm CD player is really going to limit your choices. I find some CD players called "warm" to be more like "dull". Anyway, why would you want a non-warm clinical amp?
The Complete is a nice amp for the price. I like your CD choices above. A popular choice for many in a warm CD is The Planet.
If you want more detail still on the warm side you could look at some of the Creek amps, or step up to The Complete's big brother, the YBA Integre DT.
Even though you don't include it, I would consider tweeking the warmth with your speaker cable choices. They do matter. Also adding something like the Blue Circle BC62 power cord to the amp, will bring out the warmth.