You have to pay to get the rocket back anywhere near to a barge at all. If you don't want it back you just let it fall into the ocean without spending fuel on slowing it down (and more fuel bringing that fuel along so that it is there to slow it down). You don't pay for a barge. You don't pay for the engineering work associated with bringing it back.

Dropping rockets into the ocean was what everyone* did before SpaceX came along.

If you're already paying the cost to bring it back though, it's very hard to imagine that you would have to pay to scrap it. The thing is primarily a large metal tank, and some engines made up of a bunch of metal - people pay money for the privilege of scrapping metal, not the other way around.

* Technically some other countries dropped them on deserted land, or not so deserted land in the case of China.

I have no idea, but my guess is that the cost of fuel would be an order of magnitude greater than any possible disposal costs.

Also, people pay good money for high-quality metals. The scrap value is probably greater than the logistics cost of getting it to a scrapyard.

You wouldn’t even need to scrap it. Cut it up and turn it into some mega silos for agriculture or materials suppliers. Or just set it aside and donated to the Smithsonian in a few decades…

Probably not as much as if you steer it toward the graveyard

https://en.wikipedia.org/wiki/Spacecraft_cemetery

Depending on what's in the rocket, just leaving it to decay in the environment would likely lead to contamination. Especially if you have any Hypergolic fuels lying around (explode on contact, and extremely harmful to humans / the environment). So you either pay to scrap it, or pay to clean up the site in X years.

They could probably get museums to pay them for their scrap. They would have some work to remove any confidential components before handing it over though.