Well, I recreated the dimension completely and this time it worked. I don't know what I did differently, but I did mess around with the relationships quite a bit the first time. I also limited the attributes in the second attempt to the bare minimum, though I don't think that mattered. After creating the new dimension and processing it, the full date to hierarchy matched again. I'd like to know where I went wrong the first time, but not enough to try to make it happen again. I'd much rather just do it right each time. 🙂
For anyone reading in the future, my composite keys were:
Year/Month for Month
Year/Month/DayOfMonth for Day Of Month
Year/Quarter for Quarter
Names/Values set appropriately when I needed a composite key.
Relationship was set up as: Dimension -> Full Date -> Day of Month -> Month -> Quarter -> Year