I think there is some confusion here: Raspbian lite is the headless version which comes with no GUI, and by default boots to a virtual console. LXTerminal is a GUI app that requires an X server. That and the VC are not the same thing.
So is my only option here to install a GUI and then run a terminal emulator in full screen mode?
If you want features particular to a GUI terminal (such as true color), I think so. Although the framebuffer, which supports true color, is available, there are not many apps that actually use it. This does include at least one terminal emulator,1
fbterm (that's Arch wiki documentation but it is the same software on Raspbian), but I do not think it is widely used (and having glanced at it, I don't recommend it).
- All these things are called "virtual" and/or "emulator", including the bare bones VC, for way-back historical reasons (ie., you will probably never see a "real console").