So you're saying that there was no air left in the container after it sank to the bottom? 
Nope. I'm saying that the air is more compressed but due to the false bottom or for instance: a small jug that does not allow the water to crush the jar any further resulting in hard to read displacement at the side of the jug.
That's why it's dishonest.
He's measuring the displacement of the sealed container itself, not the displacement of the contents of the container.
You do understand that the whole point of experiment was to test the displacement of a constant volume at different weights/densities, don't you? You know, like the 10 cm cube of steel vs 10 cm cube of lead that we've been talking about.
And I'm telling you that it cannot be done with a small jug of water.
But he did and, regardless of the weight, the same amount of water was displaced.
Not exactly.
What appears to be the same amount of water being displaced because the measurement is far too tiny due to the dishonesty of using the small jug.
Have a think about this.
Tell me what would happen if that same jar and weights were pushed down into a 20 foot deep tank of water?
Just in case you don't know, here's what would happen.
The first jar with the single weight in would be crushed by the mass of water. It would compress the larger volume of air inside which would mean that the volume of that jar would be reduced and displacing less water.
The other jar with the varied weights in displaced more air inside the jar so when this was pushed down it would still be crushed but the air inside being much less volume due to the dense weights inside would not compress nearly as much and would displace much more water.
This would happen and that's why the small jug is a dishonest way of doing this experiment. It is and it's as simple as that.