Unable to partition internal Toshiba drive for OS X re-installation,
Don't let that poor dinosaur extinct yet.
Now I'm sorry but I'm not sure what you are referring to with 'base system' that's taking up 350GB. Assuming that it's the recovery partition, was that already in the drive? It sounds like partition map is broken. 
If you are willing to wipe that hard drive clean, there are few things you can try.
Erase the disk using terminal in recovery mode and run verifyDisk and repairDisk. If this works then you could probably just create new osx extended partition to install osx. 
If partition map is broken, you could use external computer to repair them using third party software. 
And since you mentioned that it's a MBP from around 2012, does it have an optical bay? if your model is the ones before retina screens, there is a high chance of broken SATA cable. It is a known issue with MBP from those times, and it breaks quite often, leading to very confusing hard drive problems. If possible check if hard drive works fine on other computers and if they do, it is likely that it's your SATA cable.