You'll have to explain what you mean by "the core of Islam allows doubt", what "core" are you referring to here? We take our religion from the Quran and Hadith, and I have quoted just two Hadiths that I recall off the top of my head regarding the issue of doubt, and how it is not sinful, nor automatically looked down upon. Here's one quotation from the Quran: https://quran.com/33/10
> so I'll ask this, are you saying that it's ok for Muslims to doubt Mohammad as the prophet?
You'll have to explain exactly what you mean by doubt. A Muslim accepts that Muhammad Peace be upon him is God's final messenger. If doubts happen to cross said Muslim's mind, he's still a Muslim as per the Hadith. This is different from what I feel you're implying in that the person actively tries to take down the Islamic narrative in a way that the enemies of Islam do.
The logic is this way: someone accepts Islam because he did his due diligence and was convinced. It's not logical to start doubting everything again from scratch - it means he did not build up a proper foundation to begin with.