I just did this last night on casual. The hardest part is at the beginning. Use the default trucks since they'll come with the garage addon, don't mess with the trailers. I went north (don't forget to unlock the trucks) and then west following the paved road. If you zoom in on the map, you can see little red depth markers on the water. Use this to identify the safest places to drive. The most hazardous places are in the currents, if you get into too deep of water the current makes it very difficult to move.
Once I unlocked the garage, I got lucky and found an E-7310 on the west side of the map. This will take a load of medium logs without a trailer, and is large enough to pretty much drive anywhere on the map without getting swamped. I accidentally drove it straight through the middle of the river on the west side (not at a crossing) not realizing that the last watch point was on the other side of the water...
The E-7310 does use a LOT of fuel though, so for the two trips to the objective I put a utility trailer behind it with fuel and parts so I would have enough fuel for the second trip. It didn't have any trouble dragging the trailer along the most direct route along the south edge of the map. Alternatively, you could park a different truck near the logs with fuel and refuel there for the second trip.
If you don't get the E-7310, use the biggest truck you have, preferably the D-537 which should be able to handle a log carriage and trailer at the south crossing. Worst case, use the two starting trucks with log carriages (no trailers) and make multiple trips.