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That was a good puzzle OP, I enjoyed it.
So we have two types of door. One is 'good', the other is 'bad', in terms of containing 2 double-gold boxes instead of 1.
Your chance of picking a good door at random is 1/3.
Now, when you pick a random box from the door you just picked and blindly take a ball, if you picked a bad door, your chance of getting gold is 3/8, if you picked a good door, your chance of getting gold is 5/8.
Once you have taken the first ball, and verified that it's gold, your chances for the same box containing another gold ball are 1/2 for a bad door, and 2/3 for a good door.
Now lets get to the monty hall problem.
If you picked a bad door and switched, you get a good door. This happens 2/3 of the time. If you picked a good door and switched, you get a bad door. This happens 1/3 of the time.
However, you already opened one of the boxes in the first door you picked, and therefore you'd not only be switching from a bad door to a good door, but from a bad door where you've already picked a box which might be correct, to a good door where you pick the boxes at random again. The chances change.
Choosing to stay gives you:
Choosing to switch gives you the chances associated with a brand new door:
In this case, monty hall doesn't help. Don't switch.
