Thank you for all your replies!
Fortunately, I was able to resolve this issue and pass smog!
Here's a quick write-up on how I solved this because I noticed a lack of information on our specific F25 X3.
For context, I have a 2013 X3 xDrive35i N55 with 131k miles.
I can't guarantee this could be your issue as you could actually have a leak within your Evap system and not a faulty NVLD.
I replaced the Fuel Vapor Detection Pump, or some might call it the NVLD. Part number - 16137333301
#4 in this diagram
This part is located in the rear right passenger wheel well, right under the fuel fill port.
You have to remove the fender lining to have full access. Be aware that the clips that attach to the outer cladding are one time use only. So have replacement rivets handy.
Here's a good video explaining how to remove the rivets effectively
Part numbers for replacement rivets:
White - 07147391323
Black - 51777171004
The NVLD is located in the yellow circle. The bracket is secured using two 10mm bolts. The electrical connector was a bit tricky because you have to pull a small tab before you can push down and pull it out.
I didn't feel like spending over $200 for a brand new NVLD but luckily my local BMW recycler had a fresh 2018 Alpina B7 with low miles that they were parting out and they share the same part for only the NVLD.
On top is the part pulled out of the Alpina and the bottom is from my X3. Swapping out just the NLVD was pretty straightforward, just a couple of T15 screws holding the parts to the bracket.
The dust filter and vent hose had to be removed. Removing the vent hose was a bit challenging due to them being a one time use clamp that is quite hard to clip off.
Here's another video explaining how to remove them. I ended up replacing the clamp with a normal style hose clamp.
And that's pretty much it! I feel like the hardest part of this job was putting back the fender liner because you have to fight with it trying to get it back into the slots where the outer cladding meets.
I scanned for codes right after putting everything back together and the permanent code p4056 was still there. Also, it triggered a new code, p0450 or 194804 (BMW specific) but it was just the vehicle sensing that the NVLD was removed.
After driving for about 50 miles, the permanent code vanished and it was ready for emissions testing!
I returned the Autozone gas cap as that wasn't the problem. But I still highly suggest replacing your gas cap first as your first step in diagnosing this issue.