I figured this one out myself.
There is an option that controls how dependent projects are built under the "Projects and Solutions > Build and Run" section. I overlooked this option at first, but eventually came back around to it when I was testing each setting one at a time.
The description of the check box might be incomplete or perhaps I'm just misunderstanding it. This setting appears that it does not affect the build action of the database project, but it does affect the deploy action. When this is checked, my database project, which is not the startup project builds but does not deploy on Run/Debug. When this is unchecked, my database project builds and deploys as I expect with no additional steps required.