Sounds to me like the battery is grounded to the frame and not to the block and the starter can not get enough current. This is a VERY dangerous situation on modern EFI electronics laden systems, because it can cause other harness connections to "try" and become the ground path.
Go over your wiring carefully, and MAKE SURE you have a good heavy cable from the battery to the engine block.
(In my Ford Ranger, the ground cable has a "T" fitting in the cable. It goes from battery to frame and on to the block. On my truck, the cable got internally corroded right inside the cable next to the frame connector and would NOT crank the engine)