Fixed it!
As I'd suspected, I'd missed a vital piece of code in the MP Client Game Library. Starting from line 629 in cg_main.c, there's a list of all the various cgs.media CoreShader and GlowShader for each of the saber colors and so on. I'd missed it until now and needed to add new lines of code for my saber color.
Previously, I'd added CoreShader and GlowShader lines to the static qhandle and void UI_CacheSaberGlowGraphics sections within the ui_saber.c file. That explains why the saber colors worked in the menus. I assumed the game would pull that same information from there to render the in-game saber blades, when actually it's only used by the UI menus as the name would imply. D'oh!
Thank you for your assistance, everyone! Hopefully this helps anyone else with similar issues when splicing in new colors to the roster.
Jokile8467's post in Saber blade shows as gray squares was marked as the answer