Rather than converting your document to a different colour profile, have you tried using colour management at the print stage?
To do this, don't convert your colour profile (keep it in Adobe RGB/sRGB if you're working in RGB). Instead, once you go to print, under the Colour Management section (I forget exactly what this is called but it should be labelled something similar) you'll have the option to either let the printer manage colours or use ColorSync. Choose ColorSync and from the drop down select your Sawgrass profile. Then just choose your other settings (quality, paper type etc) as normal and click Print. That should hopefully give you better results.
Photo does have soft proofing if you want to check what your print output will look like. It's implemented as an adjustment layer (Layer - New Adjustment Layer - Soft Proofing) and will allow you to select your Sawgrass profile. Make sure it's at the top of the layer stack so it affects the entire document. And finally, don't forget to delete or disable the layer before you send to print! (It influences the image like any other adjustment layer)
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