I just got a reply over on the Steam forum. You can do it! You need to select "Show Starbases" in the gameplay section of the options menu. Then you can select a "Starbase Starter" in the designer and can save your starbase designs!!! Yay!
Okay, I'm probably missing something obvious here but what size hull starter do I need to use to make a Starbase in the Ship Designer? I've made at least 1 ship of every size but none of them show up as Starbases when trying to use the Civilization Builder. My ships all show up in their respective categories in the "Available Template" tab but when I switch over to "Starbases" from "Ships" only the official races' Starbases show up.