I ran this problem past a friend of mine who occasionally scuba dives with a team to recover sunken boats or land vehicles that have ended up in deep water.
His suggestion was that inflatable flotation bladders be used to raise the object near to the surface and then the object can be towed............while still attached to the bladders.......to the location where it needs to be moved to (if in the same body of water).
After towing it to its new location, the bladders would be deflated, lowering the object to its desired position.
He said that he can't picture in his mind how you could accomplish what you have described using a fork lift on a barge.
He did however, say that he had worked a job once where a company used a large track excavator on a big barge to lift objects out of the water after they had been floated to the surface with inflatable bladders.