Illustrator - Using an RGB transparency mask in a CMYK file

My file is CMYK, but it has a Link to an RGB PSD file as a transparency mask.

The file renders properly in illustrator (the preview), but whenever I try to export to any other format, the colors are messed up.

When I converted the RGB PSD file (transparency mask) to CMYK within Illustrator, I was able to replicate the messed-up colors.

How can I convert the RGB PSD file to CMYK without messing up all the colors?

If that's not possible, is the best option to take a screenshot of the preview, and use that instead of Exporting?

Answer

Blend modes will give very unexpected results in CMYK if you're used to working with them only in RGB.

If the purpose of the PSD is only to provide a transparency mask, why not create a PSD consisting only of the (alpha channel) luminosity values and place that instead? You could create that very quickly with Image > Calculations, using the gray values for both sources, Normal blending and setting the Result to "New Document":

This creates a new Photoshop file with a single channel of grey values.

If you just want to have a Photoshop file inside an AI clipping mask, then use Edit > Convert to Profile (not Image > Mode > CMYK) in Photoshop and convert the document to your working CMYK profile (the one that you've set up in Bridge to coordinate all of your Creative Suite applications), or to the same one that you're using in AI. Be sure to use "Perceptual" or "Relative Colorimetric" as the rendering intent in that case, otherwise you may get undesirable color shifts.