Think Doublethink
In 1984 this is a term for someone who could, knowing all the facts, still believe a falsehood to be true.
Is there a word for someone who uses a misunderstood fact to support their argument?
According to Nineteen Eighty-Four by George Orwell, doublethink is:
To know and not to know, to be conscious of complete truthfulness while telling carefully constructed lies, to hold simultaneously two opinions which cancelled out, knowing them to be contradictory and believing in both of them, to use logic against logic, to repudiate morality while laying claim to it, to believe that democracy was impossible and that the Party was the guardian of democracy, to forget whatever it was necessary to forget, then to draw it back into memory again at the moment when it was needed, and then promptly to forget it again, and above all, to apply the same process to the process itself—that was the ultimate subtlety: consciously to induce unconsciousness, and then, once again, to become unconscious of the act of hypnosis you had just performed. Even to understand the word—doublethink—involved the use of doublethink.
The power of holding two contradictory beliefs in one's mind simultaneously, and accepting both of them… To tell deliberate lies while genuinely believing in them, to forget any fact that has become inconvenient, and then, when it becomes necessary again, to draw it back from oblivion for just as long as it is needed, to deny the existence of objective reality and all the while to take account of the reality which one denies—all this is indispensably necessary. Even in using the word doublethink it is necessary to exercise doublethink. For by using the word one admits that one is tampering with reality; by a fresh act of doublethink one erases this knowledge; and so on indefinitely, with the lie always one leap ahead of the truth.
E.g. North Korean citizens, think, that they live in a free country and at the same time, they accept the dictatorship of Kim. Thus, they are unaware of the fact that they are free in captivity. This is a peculiar and real-life example of Doublethink.
Wikipedia article sum it up as: https://en.wikipedia.org/wiki/Doublethink
Doublethink is the act of simultaneously accepting two mutually contradictory beliefs as correct, often in distinct social contexts. Doublethink is related to, but differs from, hypocrisy and neutrality. Also related is cognitive dissonance1, in which contradictory beliefs cause conflict in one's mind. Doublethink is notable due to a lack of cognitive dissonance—thus the person is completely unaware of any conflict or contradiction.
If you read the same book carefully, George Orwell coined a word and a phrase to represent doublethink in simple words. They can be used in same sense.
thoughtcrime
and
two plus two equals five (2 + 2 = 5)
Other closely related phrase which became very famous during 2017 in U.S. is,
Alternative facts
Which means, "Alternative facts are not facts, but falsehoods."
i.e. we are forced to believe [something] which is not true.
This phrase is very much consider as an Orwellian2. There is a Wikipedia entry on it: https://en.wikipedia.org/wiki/Alternative_facts
1: Cognitive dissonance differs from doublethink. In cognitive dissonance, contradicting belief causes conflict in the mind, whereas, in doublethink, we learn to accept contradicting ideas.
2: Refers to George Orwell, author of the book Nineteen Eighty-Four
"Deluded person" would be the closest term I can think of to your meaning.
While this doesn't strictly specify that the false belief is due to a misunderstanding of facts, it does mean a rather dogged false belief. This would not be the clinical definition--which discounts faulty information--but a quick Google search or corpus search turns up uses for people who simply misunderstand something but are resistant to being corrected.
If you, however, you meant someone who knows a particular fact is popularly misunderstood and is taking advantage of this to promote an idea, exploiting this misunderstanding, this would be 'intellectually dishonest' and a person who employs such clever, but false reasoning could well be described as a 'sophist'.