The guy is a salesman. The slab is already cracked, cutting it now just leaves you with pieces that chip out .
If you want to, there are fillers you can install in the crack that are flexible, this would be your best bet. Try a concrete supply house and talk to them abou what to use in your climate.
I would live with the bump in the floor, it's a rental.
You could also jack the slab up with a company that installes grout fill underneath, I would not spend the $$.
However, you could pour self leveling material on top that may or may not peel off when your done.I'm just guessing you don't have a big moisture issue where you are.Again..don't spend the $$.
Rentals are there to get your best bang for your buck, fix it when you sell it and the market is good.
Other "experts" will be along soon to add to the confusion...or not, good luck.