I can't think of a way around this other than the one that you described. This would be a good one for the Ideas section!
Great, idea. Thanks! I have created an idea for this topic.
Dan, there are few routes to get the interaction you are looking for.
One option is to have the relative date filter be evaluated after table calculations are evaluated by filtering on a table calculation.
With a continuous (green) table calc version of Month on the filter shelf, you can get the quick filter and results you are looking for.
Attached is an example.
Please get in touch if you want to discuss the order of operations in Tableau.