A couple questions: is this dual layer disk meant to be played back in a DVD player or are you storing (burning) the file on it, like on a flash drive? Also do you have Compressor 4 installed? I'll try to answer all of your questions..
You can easily compress the 22GB file down to 4-5GB while retaining ~98% quality. This can be done with Compressor or even free applications like Handbrake. (Compressor 4 is recommended if you plan on doing a lot of this work). If the video is meant to be played back on a DVD in movie format then you don't have to use the 4-5GB file as your source for DVD encoding, you're better off using the full size 22GB file which will then be fit onto the disc using MPEG-2 encoding. If you plan on going this route it helps to have Compressor 4 but is not required. You can choose "DVD" as one of the Share options inside FCP X share menu. Make sure you're sharing the project containing the video meant for DVD. Then follow the on screen instructions to title your DVD, etc and click 'burn'.
Here's the Compressor route (Recommended): Drag your exported ProRes 422 file into the Compressor 4 window, and then for presets under Disc Burning drag the MPEG-2 For DVD and Dolby Digital Professional onto the movie object in Compressor to apply those settings. Click on the MPEG-2 setting and in the inspector click on "Quality". Change the quality to Two Pass VBR Best. Leave the Dolby Digital Professional alone. Then click on the gray area under the settings on your movie in Compressor and in the inspector go to "Job Action" and choose Create DVD. You can now set a Title, choose whether DVD "Plays Automatically" or "Go To Menu" when inserted. You can set a background if you choose. Submit the job and viola!
Here's how to reduce the filesize while retaining quality using Compressor 4: Drag your exported file (use Pro Res 422 as the export format) into the Compressor window. Use one of the default settings "1080p Video Sharing". Go to the inspector, click on "Settings" for video and adjust the "Automatic" quality to "Best". Go to sound "Settings" and use a high quality compressed format like AAC with Variable Bit Rate (VBR) set to "Best" quality. Then after you save these settings "Submit" the job and compressor will produce a high quality file "originalfilename-1080p Video Sharing" which you can then put on external storage and deliver to your client for his or her archives.