TD-27: Can I output the Click sound only to headphones?

Yes.
Here's how to assign the Click sound to the headphones (PHONES) output only in the TD-27: 

1. Press the [SYSTEM] button.
2. Use the cursor button to select “OUTPUT,” and then press the [ENTER] button.
3. Press the function 3 [F3] (OTHER MAS) button.
   - The "OTHER OUTPUT ASSIGN MASTER" screen appears.
4. Use the cursor buttons to highlight "CLICK," and then turn the dial counterclockwise so the indicator is selected on "PHONES" (only). 
5. When you're finished, press the [DRUM KIT] button to return to the DRUM KIT screen. The new setting is saved automatically.