Mystery Resolved (I hope). After my initial post, I ran several diagnostics each time my D3 would "die" in the middle of a shoot. I think I've found the problem. It seems to be dirty battery contacts as well as loose EN-EL4 Batteries. I noticed that a screw on the outside bottom left corner of the battery compartment is missing, therefore, causing the battery to come loose. After addressing this problem and cleaning the contacts inside the compartment and the ones on the battery, I've not had the problem again.
Hopefully, this will help others with a similar problem.