Well to be entirely fair the progression isn't entirely in one key. One thing I forgot is that it's a C major chord, and in A harmonic minor the C wouldn't be minor, like you thought, but rather augmented. This might cause the A hm to sound strange over the C chord, but only cause of that one note, and it's easy to consider the G# an avoid tone for that chord, or at least to realize that it may sound a bit out.
Again, lets spell out A harmonic minor and figure out why C is augmented (not minor) and D is minor (not major)
A B C D E F G# is the A hm scale.
7 1 3 5
The degrees I marked make up the notes in the C chord built from the A hm scale. As you can see it is an augmented 7th chord, not a minor one, but also not a major one. Since you're playing a C major triad in the progression, which contains a G, and the scale contains a G#, there might be some tension, use your ears. The thing is, that maj7 (in relation to the A scale) note is already supposed to create a lot of tension as it is often played against a m7 chord in order to make the resolution to the root note that much stronger. For this reason I'd imagine that continuing to use the G# note in your soloing over the C chord wouldn't sound strange as long as you are aware of what sound it will create and know how to use it.
What makes you say that there is a D major chord in A harmonic minor?
A B C D E F G#
5 7 1 3
F is the minor third of D, not the major third. D, F, A, and C spell out a plain old minor 7th chord. You're playing a minor triad, that also works fine, and as you can see D, F, and A are all contained in A harmonic minor.
That clear it up?