A promising option is the SeaCat between
Keelung Taipei and Pingtan. It should be priced from $2,500 NTD and take three hours.
But the website linked to from Seat 61 seems more like a squatted site full of generic Chinese or Taiwanese tourism info. I can't find any information on it about the ferry, not even the Chinese word for ferry: "渡船".
This makes me worry that the ferry may have gone out of business, and some site squatting company has taken over their page.
I tried Googling for the ferry but mainly find articles announcing it first going into service a few years ago and not much useful more recent info.
Then again the site was known not to have English and could just be badly designed. Or their old site could've got taken over by squatters even though the ferry is still in business if the company has no clue about the internet.
I worked out that the Chinese name of the ferry company is 海峡高速 (Haixia Gaosu) "Strait High-speed", which is much more amenable to Googling. I have now contributed an answer below.