Update: I figured out a solution to this problem, to an extent.
The idea is that users can navigate to my report in A from any one record in C. I want to exclude from that report all records in A which are already joined with the record from C.
The key to the solution is in the fact that when navigating from table C to A, my users always use a formula URL I have set up that navigates them to the report. I configured that formula so that in addition to directing the users to the report, it also stores the Record ID# of the record in C to a secondary table I set up for storing data.
On table A I then set up two fields:
1. A lookup field to grab the record ID# from the storage table. Let's call it CID#
2. A summary field, which displays the number of related records in the join table B where Related C is equal to CID#. This is going to be equal to 1 or 0. I named the summary field Track Existing Joins
(Note that Related C is a reference field on the join table B)
Finally, I filtered the report in A to only show records where Track Existing Joins = 0
I hope this is helpful. I recognize a lot of it is pretty specific to how I have my app setup, but I have been stuck on this problem for a month now and I wanted to help out anyone else in a similar position.