Correct, purchasing a car through MAS does not affect you HHG as all vehicles entering the country go through the DoD VPC entity. If you wanted access to your Versys sooner and you were not bringing a car, you can ship it as POV vice in your HHG. My car took around 45 days from the East Coast, my HHG is still in route at 45 days but it cannot be delivered until you find an economy residence (if unaccompanied) whenever it does arrive. Regardless, if you order a car through MAS, it may be a quick delivery or it may take some months to be built and shipped from Germany. Contact the MAS representative from Aviano to inquire about delivery times and availability.
You will need a car asap when you get here though as everything is not as convenient as in the states and getting around can be a chore without a vehicle. A rental car is an option but it will run you af least 1000€ or more per month. If have to buy a beater until your VW arrives, then that will use up one of your three registrations.
To be honest, in respect to an actual car, I would skip MAS and just buy a nice used car (automatic) in the states to ship instead (no more than 10K max). Again, Italians are terrible drivers and approx 1 in 3 Americans get in accidents during their tour here. I said accidents, however pretty much everyone gets in random fender benders, end up with random dents that were not there before they parked, or have their side mirrors taken off in traffic by mopeds/motorcycles or the random car/truck driver talking on their mobiles or texting (I have seen moped/motorcycle riders texting while riding...amazingly stupid as it sounds). This is the reason Americans drive beaters here. It is much less stressful when you don't care if a random Italian gets super close to your car in traffic and eventually makes contact or "nudges" you. If you buy a new nice, expensive car and someone hits you in a piece of sh*t Fiat with no insurance, you will be more likely to murder them and end up in the brig versus if you just had a beater instead.
Also, I say automatic because when you eventually leave, you can sell it easier if it is an automatic as most Italian vehicles are manual. Americans here suck ass at driving manuals in this mountainous terrain, so automatics fetch a higher asking price and are usually worth $1-2K more than the manual equivalent.