So laser drills because they can go lower than 6mil? The annular ring requirement is pretty strict, and the pad size cannot increase because there is no space left, so the minimum ring distance still could not be met right? I guess the tolerance on the drill head should be pretty low.
With a laser you can go smaller than 6mil. 4mil is pretty common. As long as you meet the aspect ratio requirements you should be fine. That means you may only be able to laser drill from layer 1-2. Once you are a layer below the BGA you gain space because there are no pads. You can make a small trace over to another via that goes to some other layer.
Drill pairs example 8 layer board:
1-8 through
1-2 blind
1-3 blind
2-3 buried
3-6 buried
6-7 buried
6-8 blind
7-8 blind
1_______________________
2___| |_| |________| |_____
3______| |_| |_____| |_____
4___________| |___| |_____
5___________| |___| |_____
6___________| |___| |_____
7______| |_| |_____| |_____
8___| |_| |________| |_____
You should place your via directly in the pad of the BGA. You might need to use a few layers/drill pairs to fanout the BGA properly.