Well, I went to the Genius Bar today.
They ran diagnostics and a bunch of tests on both my watch and on my phone and both had no problems.
The problem to me seems to be with the phone losing Bluetooth connectivity. Again today I had no issues until I hooked it up to my carplay unit with the lightning cable and then I watched the Bluetooth cycle on and off on the phone. I tried a test to leave my phone disconnected after resetting my watch (which we did at the apple store) and after that I had no issues yet.
I am going to try not connecting it again tomorrow during my commute and sew if I have any issues throughout the day. If not, we can safely assume my issue is caused by a software glitch with carplay and Bluetooth after the iOS 10 update.
I also have a watch series 2 coming on Wednesday. Using a different watch and getting the same result will also help to narrow down the problem to the Bluetooth glitch. I'll report back with my findings.