With your RGB document open you should just have to use

FIle >> Document Colro Mode >> CMYK

Can you select one of the white circles and check to see what blending mode may be applied in the transparency palette.

The blending modes appear different in the different color spaces because they are not affected by color management but the biggest change is in gradients. If the artwork uses one or a few gradients just redo them in the CMYK  color space and you should be fine. For best possible match in order to use color management you have to first select all and rasterize the image at the output ppi resolution. Then when  you convert to CMYK only out of gamut colors if any, will cause change in appearance.

Yes, there are some out of gamut RGB colors but they are not a problem for  the color management conversion if there is no gradients and certain blending modes. This is straight conversion from sRGB to u.s. web coated swap v2. using the raster image and it would be the same if using vector objects with flat colors. Given the narrower CMYK gamut, I think that's the best it can get in matching the original RGB.

If you select one of the now-white circles, can you "Select -> Same -> Fill color" to pick up the rest of them?  Is it possibly solvable by switching the blending modes to something that works, or adjusting the converted color to something that works with the blending mode?

Just out of curiosity, what happens if you go to View> Overprint preview?

Try opening your file in Acrobat and go to Tools> Print Production> Convert colors> Output intent (SWOP). If you are lucky, your file will be converted to CMYK without the missing parts, and will retain the vectors, although the colors will change.