The audio "WAV" file adds an interesting troubleshooting tool to this thread.
Unrelated to the re-wind issue...
It kind of sounds like a bearing noise ramps down in amplitude after the motor is shut off indicating a mechanical issue
in addition to the incorrect stator winding. If it were an electrical noise, it would disappear immediately upon cutting power.
It also sounded to me as if the motor took longer than necessary to come up to speed for the size it is.
Almost as if it was trying to start on a lower voltage. Others may chime in on what they hear and offer their interpretation of the noise.
As for finding a repair shop, here's an option:
Depending on where you are located, fill in what blanks you can, and scroll the results.
But here's the thing. Just because a business may be listed, it does not mean the repair shop is competent.
Not all shops have state-of-the-art resources and equipment.
And not all shops have people aboard who can competently develop a winding for a bare core.
So in seeking out help... it will be necessary to explain you're looking for a shop that can redesign, or reestablish a winding.
Don't rely on, "Oh sure, we can do that." Explore their oh sureness with additional questions.
Today, developing a winding for a bare core has been simplified by computer software.
Once upon a time, shops had to do all the math with a calculator, determine stator-rotor slot combinations, approximate flux densities,
and chord factors utilizing published charts etc. Computer software has streamlined the whole process now.
If you decide to get it fixed AGAIN... I wish you productive success.
John