Ok. So, first I tried to do another operation in background on a small set of data. I did an Identity with 10 polygons as input and my 860 polygons as identity features. This worked just fine. Then, I tried to run another buffer on a tiny point file with just 10 points again. This worked just fine also. I ran the buffer on my 1.8 million points again, just as I did before my last reply, and it worked swimmingly this time, executing the buffer in just over 5 minutes. The time difference from when I ran it in the foreground (24 minutes) would indicate it is processing in 64-bit background. I cannot figure out why I got the error last time. As far as I can tell, I did nothing different.
Let me know when you get the FTP site to me. Unfortunately, I am working with a dummy data set of sensitive location data kept blind by the government agency that collects and manages this data. I am packaging a model together for the agency to run for me on the real data. Nonetheless, If I can identify what sorts of shortcomings my system has that are contributing to the failure of the tool, I should be able to identify hangups for them based on their system configurations. If you are able to be successful, it will be helpful to know what differences we have that might lead to errors.