In the Key of F min C is the V. If you are trying to create a resolution (you'd be in harmonic or melodic at this point) then you'd want to use the C maj, or more appropriately C7. Otherwise moving to C minor in the key of F min is fine. You wrote it so you must make that decision. If you are avoiding the E, Eb in the first 4 bars I'd look at the melody. Is the melody in the first line hinting at melodic or harmonic minor? If not then the "ambiguity" may not be an issue. From a strict harmony point of view I am not used to this, I'd pick a 3rd and use it.
That being said, without the appearance of E in the melody and without the need for a true resolution in the first line it is fine to stick with natural minor, waiting for the next line or last line to create the tension of the V7. In line one you are not approaching the F from below so there may not be a strong feeling of resolution. I'd experiment with both Cmin and Cmaj and see what the difference is. Putting the C7 in the last measure of the first line would create the classic resolution and your melody is a G so it will work. You just need to decide whether you want that strong of a harmony there.
I would guess that the "classic" approach to harmonizing in a minor key would be to use melodic minor and that might push you into a C7 at the end of the melodic ideas (motifs).