Well, I figured it out. It is as simple as using
ScriptManager.RegisterClientScriptResource(this, typeof(ScriptTest), "Core.Resources.Scripts.ScriptTest.js");
instead of
Page.ClientScript.RegisterClientScriptResource(typeof(ScriptTest), "Core.Resources.Scripts.ScriptTest.js");
I was confused because I didn't realize RegisterClientScriptResource is a static method on the ScriptManager class. I thought I had to get a reference of the ScriptManager from the Page.