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I’m going to be in Japan for a total of 88 days with confirmed flights in and out. I was thinking about spending a week in the middle of that in South Korea, travelling on the JR Beetle from Hakata to Busan and back. The current JR Beetle site is Japanese and Korean only, and is pretty inscrutable (huge amount of options, Google Translate not making much headway). I found a ferry booking site in English that advertises that route, but their relevant port tax page has this info at the bottom:

When departing from Hakata/Fukuoka: passengers staying in Japan for more than 2 months on a tourist visa will not be able to travel on the ferry.

That seems pretty clear (I’ve contacted them about it to double-check) but does anyone have any confirmation from an official source that I wouldn’t be allowed to travel?

I made this exact trip at the end of June 2017. I saw this website before I went, and bookmarked it to post my experiences. I was traveling with my daughter, who'd been in Japan for the previous 2.5 months.

I too saw the exact warning on the Aferry website: Passengers staying in Japan for more than 2 months on a tourist visa will not be able to travel on the ferry.

It was exactly as the commenters state above.

We had no difficulty, at all, when leaving Fukuoka. No questions asked when we exchanged the Aferry vouchers for boarding passes.

For the return from Busan, we had to show printouts of our flight itinerary, to prove we had tickets to leave Japan. This was at the ferry terminal, before we received boarding passes. We did not need to prove that we had money.

Maybe worth mentioning also that the terminal and fuel taxes when leaving Busan were about half of what AFerry said they'd be. We could've spent those Won.

Customs and immigration were straightforward - actually pleasant compared to the airport experience.

Yes, you can. My email asking your question, including all of the specifics, to the ministry in Japan which oversees such, received this response:

東京入国管理局インフォメーションセンター

This is the Immigration Center at “Immigration Bureau of Japan” of “Ministry of Justice”. In regards to your inquiries, we will answer as following.

As for Immigration Bureau of Japan, as long as temporary visitors leave Japan by the expiration dates of landing permissions, there is no limitation on transportation people use.

In addition, several shared their experience on Lonely Planet.

Giora: I took a boat recently from Busan to Shimonoseki, in Japan. The boat company wouldn't let me buy a one-way Busan-Shimonoseki boat ticket without showing them proof that I had some kind of flight/boat ticket out of Japan. I suspect it's the same thing for the Busan-Fukuoka ferry. Japanese immigration didn't ask me for anything, but I wouldn't have gotten on the boat in the first place without some kind of a ticket out of Japan.

Bamse: ...[A]s a tourist, you need to have a ticket out of Japan when entering Japan (whether by flight of boat does not matter). Also as far as I know usually the airline company (or boat company in your case) and not Japanese immigration would check this requirement, basically because they are the ones risking a penalty if they let you in on a one way ticket.

As you appear to be from a country (UK) whose nationals can enter Japan visa free, and remain for 88 days, the feedback from the immigration authorities would suggest that you would want to carry with you details of your flight departure from Japan, and book a round-trip ferry passage for the trip to Korea..

On the return, according to USAToday travel writer Eric Som, Korean law requires all ferries leaving South Korea to Japan to check the immigration paperwork of all passengers.

And The Mad Traveler Kevin Revolinski has an excellent article on Taking the Ferry from Korea to Japan (Busan-Fukuoka), with everything you could possibly need to know, including the terminal GPS coordinates.