You can not fly in the US with a pressurized SCUBA tank.
According to the TSA, you can fly with an empty SCUBA tank (valve removed), however many airlines have rules that prohibit it anyway. There is no federal law I'm aware of that prohibits an unpressurized tank in the cabin, however the TSA checkpoint agents are not rocket scientists and there's a good chance they won't let you board with it. They once stopped be from flying with a set of pots and pans because "they could be used as weapons"
I see you said "privately" If it's a charter, you can take whatever they'll let you take, but still can't fly with a pressurized SCUBA tank (unless the pilot really just doesn't care)
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