The only way I can think of is to use a Spot Colour in Illustrator.
You can then place that into InDesign and the Spot Colour should show up in the Swatches Panel.
In InDesign - create a New Swatch and have this be a Spot colour too.
Using the Swatches Panel flyout menu (top right) you can go Ink Manager...
From here you can Map the colour used in Illustrator to the colour you have creaeted in InDesign.
This will replace the colour in InDesign only - and results may vary on output.
You will need to use View>Overprint Preview to view the results.
Hope that helps.
Ah, but you cannot map a spot to a process swatch, so the options are still limited. You can map the spot to black, or you can convert to process, but you cannot do both in the same file. I think a better alternative would be to make a few versions of the logo natively, spot (if it uses spot colors) process (and if the colors are important you may want to embed the profile) and grayscale. You'll have a lot more control over the grayscale in particular that way.
Thank you both for your help! In this case I used Mr. Tyson's advice and was able to implement a solution by following his directions.