Muhammad claimed (and 1.5 billion Muslims believe) that he was the FINAL prophet sent by God. Joseph Smith Jr. claimed that God ordained him to a Prophet.
How do we harmonize both claims?
The phrase used canonically in the Quran is “Seal of the Prophets” (33:40), and there’s a few hadiths that quote Muhammad saying something similar. There’s been some academic debate as to whether that means he’s the final prophet, or just that he’s the final messenger, or if it means he ratifies the truth of what was revealed to previous prophet. For most Muslims though, they take it to mean what you’ve said: there are not, and never will be, another prophet after Muhammad.
Latter-day Saints, however, claim 15 living prophets, with Joseph Smith filling a role similar to a Muslim “messenger”. How do we harmonize both? We don’t. These teachings are too different to be brought together as-is. One or both will have to be altered or discarded.
I’ve written before on the Quran, and will close with an excerpt.
While I find great virtues in the teachings of the Muslim Prophet, I’m afraid I cannot reconcile all of his teachings with our beliefs. If he was a true prophet (in the LDS sense), then either these false teachings were added later (making the Quran something like the apocrypha mentioned in D&C 91), or they were intended as a stepping-stone to move his people from the polytheism that was so prevalent at the time (in which case, the teachings are now obsolete and in need of an update). If he was not, but sincerely considered these to be true revelations, I cannot very well fault him and must leave him to God’s judgments. Finally, if he was a charlatan of the worst character, I must again leave him to God, but shudder as I do so. At its face, I find Muhammad’s legacy one of non-Christian moral virtue. We should feel no shame in partnering with such while still respectfully acknowledging our differences.