Would splicing existing wires (from charge controller to battery) to connect an inverter work, and is it safe?
May I ask what are you using the inverter for?
If you want to draw power from the battery to power anything, it wouldn't be safe to splice the wires. The charge controller takes care of the system, ensuring that the battery doesn't get over charged and goes through the proper charge cycles.
Some charge controllers will have a terminal for a load and can be programmed to power that load at certain times of the day. So if you want to power a device from the battery, go through the charge controller. I have previously used blue sky controllers.
If you are planning on splicing to have another means of charging your battery, it is definitely a bad idea. Some charge controllers have a master-slave capability, this allows you to hook up another charge controller that is a "slave" to the current charge controller and works in unison with the master to charge the battery.
So I guess my overall consensus is to stay away from splicing. If you would simply like to monitor your battery health with some measuring tool, then that would be the only case I would encourage probing the battery terminals, but still avoid splicing.
Hope that helps, let me know if you need me to clarify anything.
Good luck! 
- Josh