Color fidelity is just one of the issues you will encounter, and one that can be mostly corrected in digital post processing. In fact I have heard from people that they process C41 film in RA4 chemistry (cheap!!! ) and allegedly get decent results after scanning (obviously they don't enlarge optically).

The bigger problem that I foresee is color stability. A lot of effort has been spent on color processes to obtain long term stable colors, and anyone looking at 30 year old color prints (faded, strong color cast) can confirm that this effort was very, very necessary. These days Kodak and Fuji claim that properly C41/E6/RA4 processed materials are more or less archival for many decades.

But if you cross process, you throw all this effort away and are completely on your own. Unless your images are for immediate consumption only (read Lomo), I'd stay away from incorrect processing.