If you really want to have fun in the desert, I'd get something a bit smaller (CRF250 or 450 or a KTM, for example). Most people with smaller bikes get a trailer or a pickup truck and drive out to their favourite riding spots.
That also means you can carry spares, fuel, drinks and a change of clothes etc.
There should be no problem buying any vehicle (car or bike) in Dubai or elsewhere. Assuming you buy a registered vehicle with a road number plate, you 'export' the vehicle to your Emirate of residence. However, in the case of off-road bikes you will have to check whether Abu Dhabi emirate allows a road plate.
As an example, my car is registered in Sharjah but Sharjah does not allow off-road bikes which do not have indicators, mirror, horn etc. to be registered for the road. I got my employer to change my visa to Dubai where the registration laws are more relaxed and got a number plate.
If Abu Dhabi doesn't allow CRF-type bikes on the roads, that would be another reason to get a trailer. Or as you mentioned, get a bigger bike which is more street-orientated (but harder work in the desert).
There are usually good bikes available in Dubai at the main workshops (Gecko, Sandstorm, Motorcycle City) but try dubizzle