SSAS OLAP Semi-Additive Measure over Time Dimension
Solution
The easiest way to accomplish the above is to set the CustomRollupColumn in your date dimension for both your Quarter and Year dimensions to the following equation:
'TAIL(NonEmptyCrossJoin(
    DESCENDANTS([Date].[Date Hierarchy].CurrentMember,1),1)
).item(0)'

Where:

[Date] = Dimension Name
[Date Hierarchy] = Your custom date hierarchy dimension

Important Considerations
You cannot manually set the CustomRollupColumn equation in Visual Studio editor. This field needs to provided by a column in your dimension dataset. While this seems a little strange, it actually provides you with a ton of flexibility.
To do this, create a custom column in your DSV (or include it in your dimension table). Here is an example:

Finally, once you've completed this step navigate to your date dimension and set your CustomRollupColumn parameter equal to this newly created field. Here is an example:

Make sure you do this for both your Quarter and Yearly dimension.
Final notes
I still need to conduct some tests, but if everything works I will come back and mark this as answered. So far, everything is working great. Partially completed quarters will reference the most recent provided month (data point). Same thing with the year dimensions.
Resources that were helpful in figuring this out:

MSDN Forum Post - Get Last Value
How to Determine the First or Last Member with Data
Using the descendants function
Creating custom roll-ups Major help!
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