To answer your question, yes rabbits can get diabets insipidus both nephrogenic & pituitary types but it's rare.
Beatrix Potter's Mopsey had DI about 3 years ago.
In diabetes insipidus the urine has a very low specific gravity, which does not rise with
very cautious fluid deprivation.
I would be surprised if your bun has DI because typically, the urea should be very low/unrecordable.
I hope your vet is checking for E cuniculi, a protozoan which usually causes neurological issues but can present with kidney issues alone. It is easily treated.
The oocysts (Spores) of E.cuniculi can occasionally be seen in the sediment of urine.
diagnosis is usually by rising titers of antibodies, but they can be misleading.
I disagree very strongly with the feeding methods used by Lucille Moore. They are far removed from the natural rabbit diet. Rabbit gut micooganisms are a key part of the digestion process, & can easily become inbalanced with high concentrations of sugar & protein.
My own rabbit passed away last week aged 6 years4 months, which is an incredibly long life for a rabbit with cowpile syndrome/dysautonomia. The diet which suited him was based on a natural wild rabbit diet, was the key to his longevity, & has helped many, many bunnies.
Please listen to Graham L who has worked a miracle with his bunny Biscuit by adapting some of our findings re. bunny diet.
With every good wish for you both. Please keep us updated.