This took a lot of digging and testing.
It appears that Joomla itself (JDate) must be pulling the time from the Default timezone and not paying any attention to any time zone set int he date.timezone override. However, something strange seems to be happening with the way that this plugin is handling the renewal subscriptions. It's always doing some 'math' on the renewal subscription - adding or subtracting the difference between the Default timezone on the server and the Joomla TimeZone; however, all the times in Akeeba Subscription tables are supposed to be in UTC.
I wasn't able to track down the problematic code in this plugin, but I did come up with a work-around:
Just make sure that EVERYTHING on your server is set to UTC (Default timezone, date.timezone, Joomla global config time zone) and then this plugin doesn't see any time difference and the renewals are handled almost* properly.
* I say almost because there's a problem with multiple renewals, but I'll post that in another topic.
Hope this helps someone else that runs into this.