So I found this method from a Reddit thread, I tried it and it totally works! I will post the steps here, and if you need any help then shoot me a PM!

A VPN A credit/debit card Paypal won't work. Not required but it helps Google Chrome.

Now onto the steps.

Step 1: Add a card to your account for use on the Japanese Xbox store.
To do this go to [ Register or Signin to view external links. ]
Click "Add a payment option" at the top of the page
Select Japan and credit card
Fill out the forms. It seems like you can put anything in for the address as long as the postal code and state match. I used the address for the US embassy in Tokyo.
If it works you will see the payment option along with any other cards you have saved on your account.

Step 2: Use your VPN to give yourself a Japanese IP address.
There are a number of free VPNs available. I personally like vyprVPN.
You need to do this in order to check out on the Japanese store.
Set the location to Japan and log into your Xbox account on the site [ Register or Signin to view external links. ]
You may need to authenticate your account. If so just translate the page using chrome and follow the steps.

Step 3: Purchase your digital copy of DS3
Go to this address [ Register or Signin to view external links. ]
If this address is giving you trouble go to [ Register or Signin to view external links. ] and search "souls" you should see the game then.
and buy the game. If you did step 1 correctly you should see the form of payment you added for the region. Check out and if it worked the game should be available to download in your region.

Step 4: Download and enjoy.

The reason this works is because Xbox one games are region free. So when purchasing from Japan you have access to all other versions of the game.
Hope this helps anyone with an Xbox wanting to play early.

IF YOU DID ALL THESE STEPS AND ARE STILL GETTING THE PBR ERROR USE THE METHOD BELOW OR JUST WANT TO USE PAYPAL Props to /u/deathflex48 and /u/dutrank

Okay, to help people out who are interested in trying this and are still confused I thought I would stop by and share some advice about the Paypal method. Hopefully this clears things up for some people. As stated in the OP try this at your own risk.
A few things worth noting:
A) You DO NOT need a VPN for this method (although it's probably a good idea to have one just for general use, imo).
B) Be aware you'll be buying 8,000 yen with this method, which is more than the game itself costs. I'd only recommend going the Paypal route if you are running into a PBR error while trying to purchase the game with a credit card.

Alright on to the general directions: (I apologize if these are still a little vague, hopefully between this and the OP though it will help anyone out still having issues purchasing the game)

1. As some have already mentioned, if you're paranoid about getting your account suspended go ahead and make a fake account.
2. Once your account is set up follow the directions for adding a new payment method as mentioned in the OP, but instead connect a Paypal account instead of a credit card.
3. Now go to the Microsoft store and change the region to Japan (you'll find the option to change the region on the bottom left-hand side of the page).
4. Now you'll need to navigate to the Xbox digital gifts cards. This is a lot easier if you are using Google Chrome as a translator. To get to the gift card page I just punched into the search bar.
5. Buy 8,000 yen worth of Xbox gift cards using your Paypal account.
6. After that's done select the drop down menu in the top right corner and click on "Digital Content." You should see the codes for your gift cards here.
7. Open the Xbox store page in another tab and change the region to Japan. Open the drop down menu in the top right corner and select redeem codes.
8. Redeem your gift card codes to your account.

You should now be able to buy Dark Souls 3 using the yen you added to your account instead of a credit card. I know a lot of this is probably self-explanatory, but if you're like me and barely use the Microsoft Store (let alone an XBone) then hopefully that clears up some questions you might have.

Quick FAQ of things people have been asking
Can I get banned for doing this?
Possibly, but its unlikely in my opinion. Do this at your own risk.
What Address should I use?
I used the address for the US Embassy
1-10-5 Akasaka Minato-ku, Tokyo 107-8420 JAPAN Phone: 03-3224-5000
I keep getting redirected to the US store when I click the link.
You may need to change the region of the store manually by going here [ Register or Signin to view external links. ]
If you keep getting redirected try clearing your cookies for the xbox site.
How much did it cost
It came out to around $66 and change. It would cost a little more if your card has foreign transaction fees. Usually around 3%
Paypal works on the Japanese site, why are you doing all these steps?
You may be able to add a Paypal account for the Microsoft store but it wont work on the Xbox store so if you want to directly download from the store without having to buy currency codes the described method works.

Source: [ Register or Signin to view external links. ]