Thanks WW. I was able to get this to work by adding an OnConnect.bat file to the \Scripts folder. I have not tried the OnDisconnect.bat option, but assume it will work as well.
I did have a gotcha to add. The script would not run unless I had Profile Update set to enabled on the Meraki MX's AnyConnect client settings. I did not have to upload a profile, but it had to be enabled or else the script would not run.
I am going to put in a ticket for this as a bug because I also noticed that when I would upload an XML profile, it would be downloaded to the clients as filename.xml.xml. I am easily able to work around it by just removing the .xml extension at the end so it overwrites the desired file.