You don't want to convert the image to the paper profile before sending to the printer.
Use the soft proof to make adjustments to the image to bring back any colour saturation, loss of contrast or other effects then send the file with its original colour profile (e.g., AdobeRGB, sRGB) to the printer and use the paper profile in the print setup, being sure to select Photoshop Manages Colors and turning Printer Color Management off.
To make it easier to adjust the proofed image, before proofing, duplicate the image. Apply the proof condition to the original. Now you've got a proofed version and an unproofed version side by side and you can make adjustments to the proofed version till they look the same again (or as close as you can get).
As Andrew says, the soft proof setup has the same effect as a profile conversion but without actually changing anything in the image file.