I think the work against the air pressure should be neglected as kind of friction. As was described the gas expansion and compression processes are equal on same altitude and the closed system has efficiency 100%.
Same for different altitudes. Yes, the pressure of air drops proportionally altitude. But the balloon still goes up in case where buoyancy force more then gravity force. On certain altitude starts gas compression process into balloon using energy from expansion process. On the end of the process the balloon will have a bigger volume than the ballon was initially on the ground. However, the buoyancy force won't enough to hold the load on certain altitude and balloon will loose altitude. On lower altitude the air pressure is bigger than on high altitude. Therefore the current air pressure will compress the balloon. When the ballon will return back to the ground the volume of the ballon will be same when it was on initial stage.

the original premise was that the temperature remains constant. therefore, the change in pressure and volume is proportional.

the forces inside and outside of the balloon, are inversely proportional along the altitude scale. changes in pressure on the way down are the opposite of the way up.

What is buoyancy?
It is the difference in pressure, caused by gravity.

Buoyancy is what makes atmospheric pressure.
gravity is more intense at the ground level, which increases the density of the air.
its the same as water pressure with depth. Though the magnitude of the forces is much less, because of the differences in density.

To make buoyant, is to cause a potential in pressure.
Pressure will always try to balance itself out. greater pressure moving to areas of lower pressure.
To move pressure opposite of this flow, is to do "work".

We haven't even begun to look at changes in temperature affecting the contents of the balloon as you rise into the cold sky
That in itself will drain heat energy from the gasses within.

Thanks a lot for detailed explanations.
The first closed system is open, because need to count work against air pressure.
I calculated air pressure work for different altitude and it has same value for potential energy of load.
For example
Using this equation for air pressure P=Po*exp(-mu*g*h/RT).
T=293K,R=8.31, g=9.81, mu=0.029, Po=101325, h=1
The dP=11.8383071 PaB
Balloon's volume to raise load 1kg is 0.897585495 m^3
Balloon's volume where buoyancy force = gravity force is 0.067020711 m^3
Using equation dW=dPdV
The work difference against air pressure for altitude difference 1m is
dW=11.8383071*0.830564784=9.83248098J
The potential energy for load 1kg on altitude 1m is
PE=mgh=1*9.81*1=9.81J

Number 9.83248098J close to 9.81J.
I agree, this won't work.

Excellent!!

Now, you can see that the Buoyant Effect - is a function of Gravity !!!

What about this perpetual mobile?
Here should not be count work against air pressure:)

Provided you can keep the heat from getting to the other magnet, I see no obvious reason to disclude research along these lines.

The mistake is that you are ignoring the second law of thermodynamics. Entropy of a closed a system always increases. A closed system will not spontaneously organise itself to greate a potential gradient be it temperature  or some other potential that can ongoingly produce work.

You talk about a closed system and then introduce the concept of it exchanging energy with another system . By definition the system is no longer closed and you must redraw the system boundaries to include what it is interacting with. If you do that then the second law always applies

Thank you for answer. You're absolutely right about second thermodynamic law. I understood mistakes and I corrected the concept.
This concept is not perpetual motion machine. This concept works on difference of magnetic field of intensity. It does not violate second thermodynamic law, because an entropy stay the same. Also, it does not violate law of conservation energy, because necessary have a magnetic field. Energy should be spent to produce the magnetic field.
All details here:
https://somephysicsideas.wordpress.com/2016/02/13/entropy-engine/