Phrase Meaning – What Does “Had No Business” Mean?

Your understanding of the meaning of the sentence is exactly right. The only thing you're missing is that the sentence is stated ironically.

Irony is speech or a situation that means one thing to an audience who lacks some information, and another thing to an audience that knows better, and the audience that knows better takes some kind of delight or bitter pleasure in understanding the difference between the two interpretations.

Here's an example of "dramatic irony". In a movie, some criminals are running from the police, they crawl through a basement window into what appears to be an abandoned building, and one of them says "Whew! We're safe now." But the audience knows that the the building is actually a police station. (It's called dramatic irony because it occurs in drama.)

In your example, Mark is playing both roles simultaneously. His past self is the ignorant one, who thought he had it all figured out (that is, understood all of it). His present self sees how wrong he was. By saying it aloud, he is letting Sarah know that he has learned that in fact he did not have it all figured out. He could have just said "I thought I had it all figured out." You could understand that literally. But the ironic version carries a stronger feeling of contrition. Actually saying the false sentence as if it were true makes it sink in more strongly, and sting (Mark) a little more.

This is a common use of irony in everyday speech: saying one thing and meaning its opposite. Usually you say it to someone who will understand that you mean the opposite of the literal meaning of your words, and together you delight in the fact that they understand you correctly while someone less in-the-know would interpret your words literally. Sometimes people speak ironically intending to be misunderstood, and then take delight that they understand and others don't.

As was pointed out on that other question you linked, this is a surprisingly tricky sentence!

Let's slowly build it up.

It will take someone else.

What is "it"? Stopping Voldemort again. Implied but never directly said. As for "take", we could say "need" instead, just to be slightly clearer.

[Stopping Voldemort again] will [need] someone else.

This isn't "someone else" as in "we need a different person", but as in "we need other people prepared to do what you did, Harry". When will they be needed? "Next time" (that is, the next time that Voldemort tries to return).

[Stopping Voldemort again] will [need] someone else who is prepared to [do the same as Harry] next time.

Now, Dumbledore is saying that this someone else doesn't have to do much. Hence, it will "merely" (or "only") take someone else. This may sound like it's demeaning Harry's efforts, but it's meant to be reassuring Harry: standing up to Voldemort is not very difficult or unlikely after all, and it doesn't take someone extra-special to do it.

[Stopping Voldemort again] will merely [need] someone else who is prepared to [do the same as Harry] next time.

What, exactly, did Harry do that Dumbledore says they need other people to do (or be prepared to do)? "Fight a losing battle"--or what seems like a losing battle. (Dumbledore omits the word "like", but I'll leave it in for this one example.)

[Stopping Voldemort again] will merely [need] someone else who is prepared to fight a losing battle next time.

[Stopping Voldemort again] will merely [need] someone else who is prepared to fight what seems [like] a losing battle next time.

And now we just replace the bracketed bits with the different wording Dumbledore uses, and we have the sentence (okay, part of a sentence) that you bolded.

It will merely take someone else who is prepared to fight what seems a losing battle next time.