I think at least one source for the confusion is that while "but build they have" is not technically incorrect here, it's the kind of phrasing that would be more appropriate in opposite situation to the one you quoted in your question: Not in one which you'd expect a certain task to be done more easily, but one in which you'd expect it to be a harder.

(Also and especially because of the word but there. I mean, the author could have used and build they have just as easily. Go figure)

Consider another example, which I found after a Google search:

"I would like to think we'll be able to go up to 40 by the end of next year," he said. "That's the goal and I definitely think it is possible. Don't forget we had a running start when we founded Star with Pleiades [of Greece]. We had a big fleet of 18 or 19 vessels instantly. With Dorado we started from scratch and it was more difficult to build the pool from scratch."

But build they have. El Paso Corp's decision to enter seven OMI-owned vessels at once last year provided a big chunk. Then came the recent entries:

So, even though¹ with Dorado they had to start from scratch and despite it being more difficult, they did build whatever they were building. I would argue that usually, the specific phrasing "but build they have" comes to emphasize success despite hardships/difficulities.

^{1 Remember the confusing usage of but in your question?}

I think at least one source for the confusion is that while "but build they have" is not technically incorrect here, it's the kind of phrasing that would be more appropriate in opposite situation to the one you quoted in your question: Not in one which you'd expect a certain task to be done more easily, but one in which you'd expect it to be a harder.

(Also and especially because of the word but there. I mean, the author could have used and build they have just as easily. Go figure)

Consider another example, which I found after a Google search:

"I would like to think we'll be able to go up to 40 by the end of next year," he said. "That's the goal and I definitely think it is possible. Don't forget we had a running start when we founded Star with Pleiades [of Greece]. We had a big fleet of 18 or 19 vessels instantly. With Dorado we started from scratch and it was more difficult to build the pool from scratch."

But build they have. El Paso Corp's decision to enter seven OMI-owned vessels at once last year provided a big chunk. Then came the recent entries:

So, even though¹ with Dorado they had to start from scratch and despite it being more difficult, they did build whatever they were building. I would argue that usually, the specific phrasing "but build they have" usually comes to emphasize success despite hardships/difficulties/challenges.

^{1 Remember the confusing usage of but in your question?}