The pressure of the column of water between the two pistons will cancel out the pressure needed for the one down below to push the water up. So you are correct here.
I will have to correct you about one thing you mentioned in you question though. The pressure you'll have to overcome, should you pump from the lower pump directly is not all of the 10.9204 atm, since you already literally pump air from the surface through the air lines where the pressure is already 1 atm (neglecting the 100 mt air column-you might want to account for that). That pressure aids in your pumping. Even if these lines weren't there you would still be pumping at sea level, so either case you don't account for the atm. pressure.
The forces you will need to really over come when pumping from the surface is the friction in all the lines.
First, water at seabed pressure will also be between the sand grains in the bucket, so the water pressure does not add up to the force that will oppose your pull. This is already at equilibrium. So away go your 9800 N.
I'm not sure either about the 231 N, as the sand will probably mostly stay with the seabed as you pull out the bucket.
So your hope is in actual suction, that is, how much force is necessary to let water in when you pull the bucket out. First, as noted by User58220, this will not resist a continuous pull, as a flow of water will start as soon as there is any pull up. So unless the cylinder is denser than water and expels it slowly after a pull, it will eventually come out after some number of pulls.
If you assume that the boundary condition is really that the sand around the cylinder wall is as packed as the rest of the seabed, then you need to estimate the permeability $k$ of the sand there and from that apply D'Arcy's law to get the flow rate as a function of force, wikipedia, with the pressure drop equal to the pulling force divided by cylinder area. You're interested in $T=Ad/Q$, the time to pull out by a distance $d$ of the cylinder, as a function of the force, $T\simeq 2\mu Ahd /(kF)$. You can work out the time for it to settle back under its weight (this is true only if $d$ is small enough that the sand did not move).
What I fear is that you'll have a detachment between the packed sand and the walls of your cylinder, and flow will be much easier there. Cylinder should be very rigid and have rough walls at the scale of sand grains to try to prevent this (gluing sand on it is used in related experiments)
Best Answer
The state from $1\text{ atm}$ to $2\text{ atm}$ is normally called decompression or contraction. An equation you can use going from one state to the next is: $$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$$ Where $P$ is pressure, $V$ is volume and $T$ is temperature. Now if you want to calculate the force you have to know the surface area of what you are (de)compressing. The equation relating force and pressure is: $P=F/A$, where $A$ is surface area.
Finally, keep in mind that you probably have to take into account outside pressure as well, because this has a substantial influence.