Hi, sounds like an issue others have run into before with water making it's way into the cabin. There's things that can be done to correct that. The water being dumped into the battery compartment is normal. The water should only be draining next to the battery, specifically the rear left corner. Just to paint a picture, the battery sits on a tray, the tray sits on a bracket and below that is just the body structure with openings where the water can drain. Nothing in there would recirculate water back into the cabin unless someone drilled a hole or there's rust (doubtful).
I haven't personally had this issue and I have to say it's because I've been cleaning the foam seal and windshield where the cowling meets every so often. Like once every 3 years. What I suspect is happening is debris is collecting near the foam seal and when the cowling is removed the debris falls into the seal and then doesn't seal correctly against the windshield when put back. Then what happens is the water doesn't make it's way to the drain grate on the cowling and basically the water drains into the cabin air intake duct (located just under the windshield, behind above the battery).
I only cleaned the driver side once before but that was due to a new windshield being installed so I had to remove the wiper blade arms to get the cowling off. But I wouldn't worry about the driver side, there's no real issue there as there's nothing that can traverse water into the cabin.
One thing I would do is check to make sure the gray foam seal wedged inside the cowling is intact and then clean it. I clean it with dish soap and water and a long bristle brush. Take your time, don't rush or scrub too hard, you don't want it to rip as it's really soft and not a replaceable part. You'll end up having to buy the whole cowling piece. I then used some low pressure compressed air to dry the foam and blow out any debris that may have remained. I think when I cleaned mine I pulled the seal out just enough to clean behind it. The seal is wrapped around the cowling and I believe heat welded in on the ends. So don't try to pull it off completely. Once cleaned, I gently pushed the seal back into the cowling before installing.
The passenger side cowling is attached a specific way so make sure that is also being done correctly. The cowling slips onto the windshield into the wedge with the foam. There's a part of the passenger cowling that clicks into the driver-side cowling. Basically where the two cowlings meet you will see a tab. Make sure that clicks. Then there's those clips that secure the passenger cowling, the battery compartment cover all together. This is important as well.
Once you clean the foam seal and re-seat the cowling, you can test if water gets in by having the hood open, battery compartment lid taken off and pouring water on your windshield and see if the water is making it's way into the cabin air intake duct. You will see water pouring around the battery and that's normal.
If you still see water draining into the cabin air filter duct, then you'll need to check things over to see where the issue is. Hopefully this helps but if not, report back and we can continue diagnosing the issue.
There's a post where created a "gutter" to direct water away from the cabin air duct. It's a great idea, but let's see if we can correct the main issue before going down this path.
QX70 09-17 - 2013 FX37 Passenger Floorboard Water leak (windsheild gasket), electrical shorts and fixes (pics)