If taking PCS leave, will Sato pay for my flight to leave location if the cost is less than going to PCS location?

SATO will change your flights not pay for them. SATO is a travel agency, they don't do anything other than make a change and charge your GTC.
They will make the change for you though. You'll be charged a fee for their services. You can alter the location of where you are flying from, basically adding a leg to your flight. You'll be charged whatever the additional amount is

Based on your location your Transportation office or ITO will schedule PCS Flights. I have had Cost Price Comparison worksheets submitted to perform the action you are describing. The paperwork details that as long as your leave location will cost less than your PCS location (to fly to), you are able to adjust to that location. However, you must pay out of pocket to get back to your PCS location (as the army has already paid the same/similar amount it would cost for the PCS). The Army traditionally will cover the most efficient (cheapest) means of travel. Hope some of this helps.

You're a 51C E5?