The correct word to use in this sentence is pertinaciously.
“My brother clung PERTINACIOUSLY to his belief that Alfre Hitchwingspan was the greatest film director of all time. “
A person who is pertinacious is someone who holds tenaciously to his own beliefs or opinions. When a person has this type of attitude he/she is usually perceived as being stubborn or obstinate. Being pertinacious could also mean that someone is extremely persistent and that no amount of recourse could change his beliefs.