elusivesalvage,
For any wreck beyond the 3 mile limit, the State has no jurisdiction, and any Spanish wreck in US waters, right up to the 24 nm limit, the wreck comes under the terms of the 1902 Treaty of Friendship between the US and Spain. This Treaty has been said by the two Governments, and confirmed by the US Court of Appeals in the SeaHunt case of 2,000 to mean that the wreck is immune from unwelcome salvage claims unless the wreck has explicitly been abandoned by its owner, or the legal successors of its owner. The 1902 Treaty over-rides the normal debate of whether or not a ship was on purely non-commercial duties. The Treaty defines what it means by a Spanish ship. The Treaty covers both Government-owned and private ships, though the courts have never specifically considered the case of a privately owned ship.
I have no doubt that in the case you describe, the Warship would be considered a Spanish ship, owned by the Kingdom of Spain, and therefore covered by the treaty, irrespective of the nationality of the company that had hired it. There might be some doubt, however, about who owns the cargo. That might belong to the private party or parties who owned it at the time of sinking. I think you would find it hard to get a court to separate the ship from the cargo, in terms of a salvage claim, because it would be hard to salvage a cargo without disturbing the remains of the ship that carried it.
If I were you, I would try to get as much information about who owned the cargo, and who had hired the Warship as I could before going any further. If you can't establish clear water between the cargo and the Spanish government, then I am sure it will be considered as belonging to the Spanish Government by default.
Armed with this information, I would trace the current owners, who might be the legal successor to an insurance company, or might be the current heir(s) of the original owners. This might not be as hard as you might initially think. Then, without disclosing the exact location of the wreck, I would go and talk to the Spanish Government, who will almost certainly refer you to their legal representative, the much maligned James Goold. By then, you will probably have hired a good maritime lawyer: you are going to need one. I would not believe all the negative press about Jim Goold, if I were you: you might be pleasantly surprised. However, if you try to recover this wreck without going through these steps, and without trying to reach an agreement with Spain, you are doomed to failure, I am afraid.
The reason why I called your initial post naive is that there has been a huge amount of debate about ownership of Spanish wrecks in US waters on this forum and elsewhere, and I found it hard to believe that anybody with a real involvement with such a wreck would not be aware of it.
Again, genuinely, good luck.
Mariner