Nope, you have it wrong, or at least scrambled.
Shutter speed controls the ambient, aperture controls both flash and ambient. So if you want to manipulate ambient shutter speed is your best tool. Shutter speed is best for ambient control because flash is dumped in one brief burst, so it doesnt care if your shutter speed is 1 second or max sync. So can darken or lighten your ambient in a given scene by adjusting the shutter speed while ISO, flash power, and aperture remain the same.
However you bring about a different element. Once you hit max sync on your flash you lose that ability. I assume you understand why the max sync on your camera is limited, if not read up on the limits of two curtain shutters. After you hit max sync, then your main option would be to close down the aperture to make the entire scene darker and within your desired range. But that makes your depth of field deeper than you might like. So what would you do? Add an ND filter to bring your shutter speed down while staying in the aperture range you desire.