So does Booster know that he can't change history or is he like Zoom where he thinks he can because of minor historical inaccuracies?
I'd say neither really at the end of the day. Booster Gold does think he can change the past (due to not being a scientific expert on time travel) but the big thing about Booster Gold that makes him different is that he thinks that he could have
not been present at the moments in which he is present.
Booster Gold thinks to some degree or another that he's preserving the timeline and actually
preventing history from getting changed. He's technically right in that if he didn't act things would play out worse than they actually did but he's also arguably wrong to some degree because there was never any chance of history being altered because Booster Gold exists in the first place and already shaped the history he read about.
Zoom thinks he's actively capable of altering things. Booster Gold also does but at a certain point he goes from thinking his impact lies in altering things to thinking his impact lies in preventing alterations and the second take on things is technically more correct than the initial one even if it does miss a lot of important details.
So wait, if we were to theoretically recruit him and take an action to make a show about him would it just be destined to fail?
No. The show would not be destined to fail. Booster Gold simply could not become famous for it in a point in which he is aware of it. The show could succeed, be massively popular and potentially even be a cultural phenomenon like shows like the Brady Bunch, I Love Lucy, Gilligan's Island and The Lone Ranger are but like how most people cannot name a single actor on those shows, Booster gold himself cannot achieve any lasting fame where he'd have been able to observe that he was famous.
Additionally, if the show was such a huge success that it would be famous in the 25th century, enough for Booster to know of it, then he simply only will have been successful under a pseudonym or there would have been an error misspelling his name or something such that he wouldn't appeared to have actually changed anything. Basically, I'd have to find a reason why Booster Gold/Michael Carter specifically is not known to be incredibly famous even if he was there and stuff like making it so that "Shooter Gold/Mikhail Cartor" was a famous actor in the 20th/21st century gives me the wiggle room to do that.
That being said this type of issue is why there will likely never be a POV from a time traveler discussing the future in great detail as any information they reveal will force things to be set in stone on at least a technical level assuming the character in question isn't insane (Abrakadabra), potentially misremembering things (Booster Gold) or both.