Every technician should stock some 3n1 start relays.
If the relay does not fix it you have it for stock.
You could check the compressor to see if its shorted to ground.
You could also ohm out the compressor windings.
You could check the start relay with a volt meter to see if it's putting power to the run and start winding.
If I was in someone's home repairing this I would just go out to the truck and get a new relay to plug in and try.
Does the compressor get hot before it stops running?
If the compressor gets hot in the short time it runs before it turns off you have a problem with the compressor.
John@FreeApplianceRepair.com
www.FreeApplianceRepair.com