What does it take to independently verify whether or not President Trump called Haiti and other countries shitholes? Could this information be found through FOIA requests?
Because they didn't record them. But looking at the question's edit history, you already knew that.
These meetings are to generate agreement, consensus, or at least compromise. That isn't dependent on the process of getting there. If they can agree on a compromise today, they should be able to agree on the same compromise tomorrow. If they misunderstood what the other was saying, there'd be no point in recording that. It's not like anything they do in the meeting is legally binding. All they are doing is deciding the parameters of an agreement. The legally binding part will be the actual bill. Both sides can read the bill to establish that it is doing what they expect.
The advantage of having the meetings closed door and unrecorded is that the two sides can say what they mean without having to worry about how such statements would be regarded politically. This allows them to suggest things that may be politically unpalatable. For example, Democrats could suggest that they might trade wall funding for a path to citizenship. If that doesn't come off, then Democrats don't want to have to explain to their voters why they were willing to consider it.
The only reason to record these meetings would be to avoid this exact problem: the two sides differing on what was said in the meeting to such an extent that one side must almost certainly be lying. But if the meetings were recorded, they would be less effective at reaching consensus. Because recorded politicians talk to the microphone, not to each other.
The whole point of the meeting is to cut through the public grandstanding. Adding microphones would just get back to that. Everything would be grandstanding for the public. People would only talk in soundbites. The Democrats might think they'd like that, but it plays to Donald Trump's strengths. He's good at saying things that put the conversation on his terms. His soundbites would get more airplay than everyone else's.
Could this information be found through FOIA requests?
Of course not. A FOIA request requires that there be something to request. That could be a recording if one existed. But we already established that there are no recordings.
Even a transcript would be suspect in this particular case. Clearly if it were to differ from the Trump administration's position, it would be the fault of the deep state stenographer. If it differed from the Democrats' position, then clearly the problem would be that it was a Republican-controlled transcriber. A recording is necessary so that people can make up their own minds.
The story has been independently verified.
The Washington Post broke the original story, attributing the claim to "several [unnamed] people". The Post is a respected news paper, so if we assume that they are not lying, having multiple sources confirm a story is one way journalists verify a story.
After the Post story, various news outlets reported further verification of the story. Slate has a good overview. Sen. Dick Durbin confirmed the report independently.
Sen. Lindsey Graham basically confirmed it, but then backpedeled because of the pressure. Sen. Tim Scott stated that Graham confirmed the story to him, adding a further independent source verifying it.
Sen. David Perdue and Sen. Tom Cotton "do not recall" - later stating that they now remember and that he hasn't said it.
Trump first didn't deny it, but later did.
So we have a story by a journalist who has multiple sources. That story has then been independently verified by two Senators (Durbin and Scott) and multiple other journalists. The only person vehemently denying the story is a person who has never been concerned with the truth. His political allies backpedaled from an initial quasi-confirmation or "do not recall" to not remembering or a not very believable denial.
What you are asking about seems to be more; possibly video or audio evidence? @Brythan answer covers that. Although given how one side handles the truth, it is not clear that that would be more convincing than classical journalistic work (see eg the inaugural crowd controversy, or the denial of the "grab them" comment or various other documented (via video/audio/tweet) cases where Trump said something only to deny having said it later).