And there is serious historical thought that Mohammed, too, was fictional

Yeah, that's a ludicrous idea too. Some people seem to think that "fictional" is the null-hypothesis, to be believed by default unless there's an extraordinary amount of evidence to the contrary. This is nonsense: Fictional characters of historical influence aren't more common than real people of historical influence.

This is bias at its most obvious.

If you believe that someone made a fictional character up, and then he made thousands of near-contemporaries believe in his existence, that's a rather extraordinary hypothesis, which has a very low prior given that nobody else seems to have ever managed this feat ever.

Muhammed existed. Jesus of Nazareth existed. The evidence are overwhelmingly in their favour -- including the various bits of inelegancies and clumsinesses in their life-stories that only real-life people display, not fictional characters constructed at their time-period. And there's not a single piece of evidence that someone authored them as fictional characters.

The real problem with either question is the excessive interest in getting the "right" answer.

Right in quotes? Are we now pretending that truth has a subjective value now? This is about being less wrong, and people that assume "fiction" to be the default hypothesis, and that discount religious texts as evidence just because they are religious texts, they are more wrong than other people.

If e.g. Socrates turned out to be a fictional character invented by Plato, philosophy wouldn't care.

Socrates existed too.