As a starting point for my answer I take this comment by bobie under his question:
... my goal is to ascertain if Galileo could in his age prove that g is the same for2 bodies as different as a feather and a hammer, without cheating ...
The problem in our experiment is air. If it wasn't for the air, the experiment would show unequivocally whether a hammer and a feather fall at exactly the same acceleration caused by Earth's gravity.
OK, why air is the problem? It is a problem as there is air surrounding our objects, and this air resists their movement through it. This resistance is a force that is proportional both to the size and to the mass of the body. Now, why does this air "hang in" there? Because it cannot free fall itself, as there is more air under it, and this air below exerts pressure up not allowing air molecules surrounding the feather and the hammer fall down freely themselves.
How can we eliminate the influence of air? We need to make the air surrounding our feather and hammer fall down as freely as they do. Then it will no longer resist the movement of our objects.
Therefore all we need to do is to put the feather and the hammer in two transparent containers, and make both containers move down side by side at exactly $g$. This will remove the pressure of Earth's atmosphere working upon the air which surrounds our objects. If the floor of the containers accelerates down at $g$, the air inside the containers is allowed to fall freely.
There is another important element of the experiment. We need to assure that the feather and the hammer inside the transparent containers do not touch the walls or the floor throughout the experiment. For this purpose we should hang them on strings from the ceiling or prop them up somehow so that before we let the containers move they will be above the floor - in the middle of the container's height, say. At the moment the experiment begins, i.e. when the containers begin to accelerate down, the feather and the hammer must be immediately (with all necessary precision) released, so they can move freely inside.
All we need to do now is to allow the containers accelerate down at $g$ long enough that we can compare (is laser technique allowed?) whether either the feather or the hammer are moving with respect to the container.
If both our objects retain the same distance to the floor of their containers throughout the experiment, it means they are each free-falling at exactly the same acceleration as the containers, which is $g$.
Now, someone might object that the feather - being lighter of the two - is being "propped up" by the air that is inside the container, while for the hammer - the heavier of the two - this same air inside does not constitute the same resistance, and therefore we cannot be sure if both are really falling. The answer is: (1) we expect the hammer to fall faster of the two, therefore if it doesn't (as presumably shown through the experiment), we can assume air is not a factor here, and (2) we are not allowed to remove air from our experiment by the very conditions set by the OP, and so it will always be a suspicion that it alters the results, no matter what we do*.
Note: How do we make the containers move at uniform $g$?
1) We can use some form of the Atwood machine, one that will assure that the acceleration is really $g$. As weights used in the machine are also prone to air drag themselves, we should make them extremely heavy as compared to the total weight of the objects and the containers - heavy enough to make air drag become negligible at least throughout the experiment (i.e. for the speeds achieved by the containers).
2) To be really certain the containers are moving at exactly $g$, we should use some electronically controlled motorized machine that would assure the required smooth movement of the containers.
*I have my doubts that it is at all possible to eliminate the microscopic air movement inside the containers. Even if we make them absolutely air-tight, there is still a chance that the initial sudden movement of the containers down might cause some turbulences. Still, if we are not allowed to evacuate the containers, any proposed experiment will have to account for that.