I just spent 10 extra days coming back from California to Georgia because of a similar problem with a 330 CAT and it was the fuel pump.
They put a new injection pump and fuel pump on in LA and worked on it again in Phoenix and put another new fuel pump on in Albuquerque, NM and it has been great since CAT replaced the fuel pump the second time.
We were towed three times at a very high cost. Fortunetly we had insurance and Freightliner and CAT did the repairs under warrant and paid to have it towed one time.
On our unit it has no primer pump. If your has a primer pump you may start it that way.