Would there be an expected evolutionary benefit for Ad-36 to increase obesity?

I don't know. But the benefit, if it exists, might be to the virus or to the human.

How can we distinguish correlation and causation in this case?

It would be easy, if medical ethics permitted the experiment of exposing half the test subjects to the virus, without telling the test subjects that this is an obesity experiment.

Have we or can we try this on experimental animal models?

I believe there are animal model test results out there. You can easily find them by Googling, etc.