After much searching across the internet I have found that you could not set the name of the file you were attaching to dynamic report subscriptions in Microsoft Reporting Services. I did manage to find one article showing you how your could make a stored procedure to actually execute the report from, providing the parameters. However this only worked for one parameter, and was kind of a pain for a long term solution.
I found the core of this method was using the path in the catalog table. That is what the file name is based off of. So I have created the stored procedure below to be run every 15 minutes (you can change the timing) the change the path of the reports that were going to be generated in the next 15 minutes and to change back the path of any reports that were previously generated.
The only limitation of this is that you cannot access that report directly in the report server during that 15 minutes. However, you can alter the script to change this timeframe if you would like to limit that window.
P.S. I have only tested this with Data Driven report subscriptions fully. This is more useful for email delivery as filebased you can just use parameters such as @TimeStamp for the filename.