Use thinset! Thinset will work fine on both the cbu and the drywall and it's a lot less expensive than mastic (regardless of what it is labeled, if it comes premixed and in a bucket, it's technically mastic, and not thinset - thinset's strength comes from cement and there's none in mastic). Depending on the size of the tiles, mastic won't work as it needs to dry and it will take months through the small joints from the edge in. Plus, as noted, if you ever get it wet, it will re-emulsify. I've taken stuff fully dried that was months old and washed it off. Took a little scrubbing, but it got soft and came off. That won't happen with thinset!
Read the installation instructions on Ditra. It requires a modified thinset to hold it down to the plywood, and then unmodified to hold the tile down. The easier way to do this is to buy the unmodified (dryset) thinset at Lowes, and for underneath, use their add-in admix (modifier) rather than the water. Then, when done and setting the tile, just mix it with water instead...same stuff, two different uses based on what you mix it with. An alternative is to use Ditraset, which Schluter has made specifically to their specifications. You don't want the cheapest dryset thinset...they have the absolute minimum amount of cement in them. HD only carries a really cheap dryset, but Lowes (and any good tile store) will have others. If you want to buy the dry thinset with the modifiers mixed in, HD's Versabond is decent. When mixing a dry modified thinset, it is essential to follow the directions and remix after waiting the full slaking time. WHen you first mix it, you wet the modifiers. It takes awhile for them to absorb the moisture. That happens during the slaking period, then you mix it again which now coats all of the cement particles. Don't do this, and it's nowhere near as strong, nor does it spread as nice. When mixing the thinset to bed the Ditra, it needs to be smooth and able to hold a notch, but you don't want it stiff and dry. After setting the Ditra in the thinset, you should be able to pull up a corner and see that all of the mesh is covered. If not, then you probably didn't either mix it well enough, or it's too stiff.