Bleeding a GM power steering pump

I've got a 1988 GM power steering pump connected to a 1995 Chrysler rack & pinion unit, it all works very well except for the pump is whining a little.

I've tried jacking it up and running the engine at idle while turning the wheel slowly left-right multiple times. I've also hooked a vacuum pump to the PS pump while turning the wheel left to right. The lines are new as is the pump and rack, I had the same problem with a used pump and rack. There is no foam in the fluid and I rarely get any bubbles during bleeding. I'm certain there are no leaks or any way for air to get in the system. The whining is not too bad, not like some I've heard.

It's not the belt, there's a big difference between the sound of a squealing belt and a whining pump!

What am I missing here, should I adapt the proper Chrysler pump to my SBC, would this solve my problem?

Not sure where the noise could be coming from or why. I can tell you that I don't think it's the 2 different parts. I have a Saginaw pump which is basically GM and it's hooked to a Ford MS2 R&P with no problems.
The problem I ran into was pump pressure. My original Saginaw pump was rated at 800 PSI................the MS2 R&P needed a 1200 PSI pump.
That may be the problem with your setup...........pump pressure.

Not sure if I read what you are doing correctly, but here is the way I would describe the breeding process. Car on the ground, pretty sure that makes no difference, how I start. Motor running turn the wheels lock too lock make sure you stop and hold some pressure when the turn limit is reach. Full left hold some pressure against the lock back off a bit repeat, then full right repeat. Go full L & R a couple of times, repeat the function described hope this is clear. That will clear the air out a whining pump you are correct usually means air.

I decided to adapt the correct Mopar pump for the rack. Two trips to the salvage to get a pump/pulley and an extra GM Alternator/Power steering mount and I spent the whole weekend adapting the Mopar pump to the GM mount. I drove the truck later that day and the GM pump had stopped whining, go figure it's been a problem for a year!

I still plan to swap the GM for the Mopar pump.

The pump will "bleed" itself, it's the box that needs bleeding. If a pump has been run out of fluid, it will often start to whine, and there's really nothing to do to stop it but change pumps.