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Technically, if the Qatari plane gets instated as property of Air Force One operations then it belongs to the US and not the president. It would have been better to lean on the fact that Saudi Arabia is a direct sponsor to many groups and nations that the USA lists as adversaries.
Given that Trump is explicitly granting it to his Presidential library or whatever bullshit, and therefore restricting its use to his own purposes - no, it would effectively not belong to the US.
Trump’s Library is vaporware. He made a lot of noise about how great it was going to be and “the best Presidential Library ever,” and grifted and conned and got donations. Still waiting for that first golden shovel to break ground.