Your theory about acquiring a disk copy of the game and not installing the updates is correct and will allow you to unlock the Grease Monkey trophy. The following is what I did:
Copy Rocket League save data to an external drive or upload it.
Disconnect the network connection.
Delete digital copy of game along with updates, and save file.
Restart PS4 to clear cache.
Insert the disc copy of the game and play one exhibition game in order to unlock a topper.
Go to the garage and equip every slot. Trophy should unlock.
Reconnect network connection, re-download game, and restore save data.
Hope this is of some help to you or anyone else attempting to unlock the Grease Monkey trophy.