I don't think he made it up. I do think he jumped the gun on talking about it which, while he's not wrong for reporting information he gets, could very well bite him in the ass later.
A letter of intent is not a guarantee that anything will come. It is little more than an understanding that they would want to work together on a project. At most, the letter might spell out who Kojima could shop the game out to besides Microsoft and when. Also, any potential penalties if one party or the other decides to not go through with anything. But without seeing the contents of the letter, we really have no idea what agreements were made. The letter doesn't even mean that exclusivity is part of the plan. It could be but not a guarantee. It also means that no definitive publishing agreement was reached at the time and likely that there wasn't even much of a project ready to get started. It was likely still just an idea.
Grubb reporting on a letter of intent is setting himself up to look like he was full of shit. It may be true, but if it falls apart, and a letter of intent very easily can, he's not going to be able to prove that it ever existed unless he gets a copy for himself that he can verify is authentic or if the ones involved admit that something was planned. He's gambling with this. Because if it happens, Grubb will definitely be elevated among "insiders". If it doesn't happen, what is true won't matter because he won't be able to prove it himself and his reputation will take a hit.
Either way, even if a project between Microsoft and Kojima exists we're not likely to hear about it for years yet. With Death Stranding 2 looking more and more real, and not even announced yet, if his Microsoft project exists and has moved forward then it's probably still far too early to reveal it. It will probably also be muzzled by the need to focus on the bigger, more immediate game for a company already paying him an undoubtedly large sum of money for it.