Using Time Machine with 2 internal drives
I agree that TimeMachine will backup both internal volumes.
https://www.baligu.com/pondini/TM/32.html

My sister's 2012 MacBook is running terribly slow

I would spend a little time diagnosing why the machine is running slow. 


Run activity monitor /Applications/Utilities/Activity Monitor.  Check ram usage and paging.
Run etrecheck https://itunes.apple.com/us/app/etrecheck/id1423715984?mt=12