There was a now banned member, Audi Junkie, who posted a number of UOAs of VAG turbo engines that had been run 5-7K on API spec synthetic thirty grades that had no claim to any VW spec.
These oils did just fine.
This member posted a lot of very useful information but also became quite irate with anyone who disagreed with him.
If an API 5W-30 synthetic will work in a turbo VW on moderate drains, it should work just fine in a 2.0.
If you really want to run a forty, I'd recommend Maxlife 10W-40.
It works great in my old BMW, which has considerably higher specific output that does a VW 2.0.
I don't think you need to go to the expense of a 5W-40, which will be at least a Grp III oil.
OTOH, if you decide to run a 5W-30, you can always find some kind of a deal on some brand of Grp III 5W-30, while 5W-40 is rarely less than $5.00/qt.
If you're having to add oil, a cheaper oil that will work well only makes sense.