It would be good to be able to use the photograph of your choice and be able to insert text that is your logo and contact details. This would ensure that the desired quality is achieved. Other software can have very disappointing results as far as colour production.
Most printers that print to CD/DVD come with software that uses any image (rectangular or square) that you position over a circular guide. I do not think Adobe would include this software in Lightroom to cater for every type of printer made.
In Lightroom you could achieve a 'Preview' or Export a JPG with your text or logo -
1. In Develop show your preferred circular area of the image by using a Radial Filter or Vignette.
2. In the Print module Add an Identity plate and/or Watermark to the photo.
3. Save the Print as a Template.
4. Export the jpg to use in the printer software.