Alright, I've think I've figured out exactly what the problem is.
Before I installed Entrenchment, I made sure that both vanilla Sins and Entrenchment were completely removed from my computer. I uninstalled them from Impulse, and then removed the Stardock Games directory in Program Files(x86) from my computer. I also removed the AppData\Local\Ironclad Games\ directory.
The game (Entrenchment Beta 3) does install and run correctly. If I verify the installation however, Impulse will tell me that it is not installed correctly, and prompt me to update the game. When I do so, it downloads a 114kb file, causing Entrenchment to no longer work with an error message stating:
"Entrenchment has detected files from a previous beta installation. Please uninstall Sins and Entrechment via Impulse and reinstall both. This is a limitation from earlier betas and will not be an issue with the Entrenchment release."
I hope this information helps to resolve this problem.