PeterDonis said:
I'm not aware of any such model. The only model of a "box of radiation" that I'm aware of has ##p = \rho / 3##, where ##\rho## is the energy density. Do you have a reference for the model of cavity radiation you are describing?
Steve Carlip has a paper relevant to the "light in a box" problem. See for instance
https://arxiv.org/abs/gr-qc/9909014, "Kinetic energy an the equivalence principle".
According to the general theory of relativity, kinetic energy contributes to gravitational mass. Surprisingly, the observational evidence for this prediction does not seem to be discussed in the literature. I reanalyze existing experimental data to test the equivalence principle for the kinetic energy of atomic electrons, and show that fairly strong limits on possible violations can be obtained. I discuss the relationship of this result to the occasional claim that ``light falls with twice the acceleration of ordinary matter.''
Note that the majority of paper is about the effect of kinetic energy of massive particles, but Carlip also discusses the "light in a box" problem.
Also relevant is a paradox, initially introduced by Tolman. See Misner et al, "Active Gravitational Mass",
https://journals.aps.org/pr/abstract/10.1103/PhysRev.116.1045. A quote from the abstract:
Tolman states that "...disordered radiation in the interior of a fluid sphere contributes roughly speaking twice as much to the gravitational field of the sphere as the same amount of energy in the form of matter." The gravitational pull exerted by a system on a distant test particle might therefore at first sight be expected to increase if within the system a pair of oppositely charged electrons annihilate to produce radiation. This apparent paradox is analyzed here in the case where gravitational effects internal to the system are unimportant. It is shown that tensions in the wall of the container compensate the effect mentioned by Tolman so that the net gravitational pull exerted by the system does not change.
To recap, if you have a very strong box, and inside the box you convert matter to radiation, the mass of the box doesn't change as a result of the conversion, as one would expect. There are different schemes for accounting for the mass - if we use the Komar scheme, we integrate ##\rho + 3P## over the box and the walls. The pressure term is positive in the interior of the box, and negative in the walls of the box, for a net result of zero for the box+wall system.
This is in principle experimentally testable - one can in principle measure the proper acceleration of a stationary test particle just inside the walls of the box and find that it's different from the proper acceleration of a test particle just outside the walls of the box. I've played with this in the past using the Schwarzschild metric - being unwilling to deal with junction conditions, I took the approach of making the model of the "walls" of the box of finite thickness so I didn't have to deal with learning about how to handle the junction conditions. I could try and dig out the posts where I did this with search, but I probably won't bother unless someone is interested enough to ask. I've long since forgotten the details.
Using the Newtonian ideas, if we have a massless box containing matter, and then convert the matter into radiation, we find the total mass of the system of box + contents unchanged, but the mass of the interior of the box (excluding the walls) is higher than we expect, in the relativistic limit the mass of the interior is doubled, the scale reading of a test mass is double E/c^2. But the scale reading outside the walls of the box is E/c^2 as we expect.
Of course the actual analysis does not use this Newtonian language. If you get past the paywall, you can read Misner's paper for his approach. The approach that I used when I was thinking about this issue was to use the Komar mass in the frame of a static observer, basically the integral of ##\rho + 3P##, and a spherically symmetrical static metric, basically the Schwarzschild interior metric.
Going back to the light in the box problem for a bit - we can consider the walls + light to be an isolated system, and use E^2 - p^2 (in relativistic units where c=1) to find m^2, E being energy, p momentum, and m mass. However, if we try to find the mass of "just the light" and exclude the mass of the walls of the box, we find that the quantity is not covariant. If one checks Taylor's "space-time physics", one sees the fine print for E^2 - p^2 gives us the mass of an "isolated system", however, while the box + light is an isolated system, the light by itself is not isolated.