I thought of this argument before and it looked plausible to me. However, I think this is wrong. The energy equation is $\frac{\partial u}{\partial t}+\nabla\cdot \mathbf S=0$ which you can used as you do here to derive the change in energy in the volume is zero. However, the momentum change equation is the equation (1) in @MariusMatutiae's answer. You need to integrate the Poynting vector over the volume not the boundary.

Were this argument true, there would have been no need for the close surface integral to arrive at zero. This would have implied wrongly the momentum transfer or pressure from a beam of light is zero for any open totally reflective plane surface like a mirror. The Poynting vector at the surface is zero precisely because the perpendicular momentum of incoming wave and outgoing wave are opposite of each other. The momentum imparted locally to the surface actually double the perpendicular incoming momentum rather than zero.

@Hans, I agree with your first comment, using Maxwell tensor is a more direct way to address your question. But I think even with Poynting energy one may suceed. I've edited my answer to clarify.

I do not see there is anything of substance changed in your answer. You are still writing about the flux of Poynting vector across a surface. In the first paragraph you recently added, you are now talking about energy, supposedly acknowledging the equation in my first objection comment. You know it is not my question. I am asking for momentum. Supposedly you are addressing the momentum issue by saying "therefore momentum of the material cavity is constant in time as well". How do you derive that? (to be continued in the next comment)

(continue from previous) Energy could be transformed into not macroscopic kinetic energy but thermal energy of the charged particle or simply EM wave inside the material. Also you state without proof "these forces cancel out and the cavity does not move". I am asking for a rigorous mathematical derivation not some handwaving lazy argument, which anybody could do without any help.

I have edited my question, to specifically ask for a mathematical proof of the volume integral of the Poynting vector or its time average being time constant. I have no need for handwaving argument.

"Energy could be transformed into not macroscopic kinetic energy but thermal energy of the charged particle or simply EM wave inside the material." No. There is no energy transferred to the matter of the cavity (the text in small font explains why). This means kinetic energy of the cavity remains the same. "thermal energy of charged particle" does not belong here, as you posed your question for a perfect conductor.

Rather than debate what kind of energy is transferred to where this moment -- I will think about it later -- are you able to prove rigorously mathematically $\frac{\partial \int\mathbf S dV}{\partial t} = 0$?

Aside from the question from my last comment, are you going to explicate mathematically your statement "if the cavity was changing its momentum during some time interval, it would be changing its kinetic energy too"? What is field momentum as a function of field energy? I have been waiting for a direct and explicit answer. I would very much appreciate it if you would state explicitly whether you will need time to answer or you are not able to answer in explicit and mathematical terms.

I think I could prove your statement, but I do not have the will to write down the proof for you at this time. I think you can try to do it yourself, by using similar reasoning that I used in the explanation of why $\oint \mathbf S\cdot d\boldsymbol \Sigma$ vanishes. Just use the Maxwell stress tensor instead of $\mathbf S$.

Regarding the cavity energy: by cavity I meant only the material body with positive mass $m$, not the EM field inside. If momentum of this cavity $\mathbf p=m\mathbf v$ is changing in time, in the rest frame of the cavity its kinetic energy has to change as well, because kinetic energy is a function of momentum: $E_k = \frac{p^2}{2m}$.

I understand perfectly well why $\oint S\cdot d\Sigma$ vanishes long before without you telling me. It was me who pointed out the confusion you had over the difference between the energy and momentum. Do you not think it is quite highhanded and condescending to say "by using similar reasoning that I used in the explanation of why... vanishes" and "just use the Maxwell stress tensor instead of S"? It was me who pointed out YOUR mistake and that YOU have to use Maxwell stress energy tensor instead of S. Don't you remember?

It is as if you are Fermat, and saying "my book margin is too small for my brilliant proof. You go ahead and try to prove my theorem and along the way the Taniyama–Shimura–Weil conjecture. It is easy for me and I will leave it as an exercise for you of low dull smarts." If you do not have the capacity to provide the proof which is what I am asking for, please have the courage and honesty acknowledging it, instead of assuming an attitude and arrogance of a master without the substance and intellect.

Regarding your comment regarding cavity energy. I am asking about the relation between cavity space field energy and field momentum, NOT the kinetic energy and momentum of the conductor enclosing the cavity. They are two distinct concepts. More importantly, I have already specified unequivocally in the question not to use conservation of momentum between the cavity and its enclosing material but the field property itself. Could you please be honest and courageous, and not beat the bush and circumvent the real issue and try to look smart and smug brandishing high school physics?

Hans, I am afraid I do not like your attitude. On my side, the expected mode of communication is to be glad for any relevant information one may get. You keep complaining and throwing adjectives at me for not providing you with the answer you like. I can say only this: I am aware the approach I explained is not perfect, but I like it because it is easy on math and uses physical reasoning. If you do not find it useful for you, my advice is to seek better answer elsewhere or figure one out yourself.

It is ironic that you should be complaining about my attitude. You do not see that your attitude has been condescending and arrogant? What would you feel if someone tells you that your question is too easy for them that it is below them to give you the right answer and you should go figure it yourself, while it is clear they have no idea how to do it themselves? On top of that they keep telling you things that are obvious but irrelevant, acting as if these are the gem of ideas. Honestly tell me, are you not going to feel insulted?

I already said in unequivocal terms in my question above that I know from physical argument with conservation of momentum that the momentum volume integral has to be constant, I am asking for mathematical reasoning from field property. And I explicitly ask you to tell me honestly whether you could do that. I will accept it if you honestly and direct acknowledge that you are not able to do it instead of equivocating "do not have the will to write down the proof for you at this time". I put the question on physics.stackexchange, it is not for you ask me to seek better answer on other website.

Instead, my advice to you is to acknowledge you are incapable of providing an answer and not pretending otherwise. You do not have to answer this question if you are not able to.

I'm moving this discussion to chat.