Is there perhaps a technique I can use to lower the pitch of the E to Eb while blowing a C triad?
No there isn't. For minor key chord stuff people tend to play chromatic harmonicas in D minor or Eb minor (and in that sense they become partially diatonic instruments). I saw a video the other day that I now can't find, of Jason Ricci playing a chromatic live (rare!), and he played a d minor vamp, using a lot of d minor 6 chords. That is to say, he approached the instrument (as you would expect), in a diatonic fashion.
Chromatic harmonicas are fully chromatic, but only for single note playing. In addition to that, the note layout does favour some keys over others. Interestingly enough, the easiest keys are those with a closest relationship to the "key" of the chromatic, OR the key a semitone up, so on a C chromatic C sharp and G sharp are exceptionally easy keys which is unusual to say the least!
The chromatic harmonica wasn't designed from the ground up as a chromatic instrument, but rather an existing diatonic layout (used to play major melodies) was used, and a slide mechanism added on top of that. This means that what you have, in effect, is a solo tuned C major diatonic harmonica that also shifts up a semitone (or, if you like, a Db major one that also shifts down). In a way though, that's part of its charm: each key has its own "feeling", much like on many other modern chromatic wind instruments, or indeed the piano for that matter.
Some players prefer to play with the chromatic tuned in diminished triads, as this makes it more "key neutral". Other think this adds very little, and loses you the chords and double stops that you do have: it's a matter of preference.
For further reading I suggest you check out what Pat Missin and Brendan power have to say about chromatic harmonica tunings, their benefits and their limitations. Also, if you have a chromatic harmonica at your disposal, make sure you learn all of Stevie Wonder's repertoire. But I guess that goes without saying.