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FWIW, I have an older version CDE-HD148BT and the door closes without issue in my MY00. Looks like a similar faceplate with the button and volume knobs having swapped places.
I have the Alpine SXM145BT, it fits fine and the door works. If you check the dimensions of the 165BT --- it's bigger, in all dimensions. You may have problems. Maybe call Crutchfield, first?
I have a CDE-153BT (will be upgrading over the winter) and have no issues with door clearance. I even have an iPhone cord plugged in the front, and it will still close.