There is nothing like a perfect world. All models have advantages and disadvantages 

Currently, in the large animal field, Ossabaw pigs are a species that have a genotype predisposing them to many symptoms associated with type II diabetes, which is exacerbated by high caloric/fat diet. There are also several mouse models available for specific phenotypes associated with metabolic diseases. Additional details would help in identifying the best fit-model. 

I have attached a PDF for your reference.

I hope that you'll find answer at following papers .

Regards

Article Animal models in diabetes research

Article The Use of Animal Models in the Study of Diabetes Mellitus

Article The use of animal models in diabetes research

I think another thing to consider is laboratory compatibility and animal availability. For example, chinchillas and degus will develop diabetes easily, but how compatible are these animals for research when you consider housing, handling, feeding, breeding, costs, etc. Some cats are predisposed to diabetes in old age, but you have to wait a decade or more for it to develop in the feline, making it a poor animal model. Also, a desert sand-rat may be a good model theoretically, but not easily available.  That narrows things down to your more traditional laboratory research rodent models (mice and rats) that develop the condition due to a combination of environmental factors and genetic predisposition, and which most research facilities are equipped to maintain.

I thank you all for the insightful responses.