Hi Derek,
The validity of any advice you receive, will depend on the country/legal administration of your aircraft.
"Odyssey PC680 12V battery in it, but I am reading that this battery is not approved for the airplane."
Do you happen to know why not?
"Ducati also says NOT to use an Odyssey with their VR"
While Ducati is the manufacturer the Rotax VR, it seem to me you should be talking to Rotax, as their technical advice pertains to aircraft use of the VR.
"Does anybody have any suggestions for a proper battery for the 912ULS engine that my airplane has?"
It is unlikly that a specific brand is the "proper battery". If you look in the Rotax Installation Manual, Chapter 24-00-00, Electrical Power. Pages 3-4 you will find the Rotax minimum battery performance specifications.
NOTE: Rotax do not specify & brand of battery.
In my last 912ULS powered aircraft, I used motorcycle (small) AGM batteries - a lot cheaper than dedicated aircarft types (eg Odyssey) with more CCA's (for much better cranking), in the same external dimensions and usually a little lighter in weight. Mine lasted 5-7 years, before being retired to mower/electric fence duties.
My current 912 ULS powered aircraft, uses an Earth X 900, aircraft certified LIFePo4 (Lithium iron phosphate). Much higher price than above, lighter, great cranking and claimed to last 10 years.
Your aircraft builder may have specified a certain Brand & model of battery - I doubt if you realy need to adhere to this, but get some advise from a knowledgeable local technician.😈