Yes, split them.
The concept of BSP trees is almost as old as it's possible for something to be, in terms of graphics programming. Their use dates back to the Quake engine in 1996, and the Quake engine and it'sits tools provide a sample implementation that has since been released under the GNU GPL, although you should be aware that the code isn't quite up to modern standards.
Nonetheless,
- Quake engine source code: https://github.com/id-Software/Quake
- Quake tools source code: https://github.com/id-Software/Quake-Tools
Michael Abrash's Graphics Programming Black Book also provides some conceptual discussion of BSP trees in it'sits later chapters, as well as some sample code of more generic implementations, and this is also available for free online in multiple places, for example: http://www.jagregory.com/abrash-black-book/ (look from Chapter 59 onwards).