One of the new features in GP 2013 R2 is the Microsoft ribbon command layout that we are familiar with in Microsoft Office products. It's kinda cool that this familiar interface is now part of Dynamics GP, even though it takes up more real estate in the window and takes some time to get used to as opposed to the previous icon shapes and positioning.
However, the Ribbon button layout wreaks havoc with Integration Manager (IM). Because IM is at heart a collection of 'macros on steroids', essentially mimicking a user with really fast typing skills, the new button layout doesn't seem to conform completely with the IM Destination Adapters. The temporary resolution is to change the setting for the window layout in a new area in User Preferences called the 'Window Command Bar'. The default setting is 'Action Pane'. You need to set it to 'Menu Bar' to make IM play nicely with GP.
After changing the setting, log out and back in to GP for the setting change to take affect.
The errors I was seeing would not have led one to believe this could be the issue. I was getting two errors, regardless of which integration I tried - 'RPC Server Error' and 'Too many Forms Open'. However, these errors didn't occur until approximately 1,000 records had been successfully imported, then the integration would fail at the same point every time. I initially suspected a data error or network time-out of some sort. After MANY hours of troubleshooting, I finally opened a support case with Microsoft and my friend Chris Bulson at Microsoft had the resolution to me within minutes.
This is a known bug that Microsoft is working on the fix for. Hopefully this article prevents someone else needless hours wasted and head-scratching.