It seems that your "knowledge" doesn't match that of the Apollo flight journal...
As usual, the concepts at work are not as simple as you wish. Yes, there is a mode in which the LM autopilot can make use of moment-of-inertia data to estimate acceleration rates. And because the DAP has to be able to fly the LM/CSM docked, undocked, and separated, these are grouped into different control laws for each case, not all of which need the moment of inertia. Ask yourself why the lunar module pilot would give a flying frack about the mass of the CSM: it's because they're setting up for the contingency in which the LM might have to fly the whole stack, not because knowing these masses is essential to solving the guidance problem.
Because the LM RCS has to be able to fly the whole stack (very massive) and the ascent stage alone (something like two orders of magnitude less massive), the control axes are not purely orthogonal, the jet firing logic is not straightforward, and the jets are oversized. It is the need to accommodate these highly varying spacecraft masses and flight regimes that requires a separate step of estimating rotation rates, angular acceleration, and thus whether to fire the jets continuously or "pulse" them. But this is not a requirement of the attitude control problem. It is an artifact of how the LM designers chose to solve it.
You bring up weight-and-balance charts, which are critical for determining the center of gravity for a winged airplane. You compute the moment arms of passengers and baggage primarily in the pitch axis (but perhaps also in the roll axis) and position them so that the center of gravity thus computed lies within a particular envelope that permits flight control. That relates to spacecraft design only insofar as the gross center of mass and the gross placement of reaction controls should be coordinated in the design. In most practical spacecraft designs, stability cannot depend on real time control or knowledge of the center of mass. It will simply be where it is.
In the case of a docked LM/CSM, the combined center of mass doesn't lie anywhere close to the ideal position for the LM RCS. Control is still possible. This problem was solved by a mathematical indirection whereby a notional set of control axes was devised, and the practical control axes were mathematically mapped to them via linear algebra depending on which flight mode was in force and which control laws governed. Again, this is not endemic to the problem of attitude control. It is merely the way Apollo decided to solve it.
The values being passed to the lunar module in the transcript are merely the spacecraft masses, not information that has anything directly to do with center of mass. We want the moment of inertia
in one case because we want to estimate our angular acceleration and decide whether to fire the jets continuously or pulse them. You can see the LM ascent stage operating in "pulse mode" in the ascent film. The moment of inertia changes extremely dramatically over the mission. This doesn't affect the essential nature of the guidance problem, but it relates to one possible way to solve it.
In general, if you can measure the error and error rate (which any IMU in the 1960s could do), then you have the basis for a proportional-differential controller in that channel, which is at the heart of MIT's design. There is no need to estimate or precompute the application of a corrective moment, and in fact the accuracy of the solution isn't increased by doing so. If the error and error rate are cross-signed, you merely have to zero out the error rate when the error falls inside the deadband. You don't need to time anything; you just apply the control until the rate reaches zero, however long that takes. You might know this as closed-loop control. If the error and error rate are same-signed, you need to apply a control moment to oppose the rate and drive it to a preset corrective value in the opposite direction. Then the previous logic takes over. Do this separately for the roll, pitch, and yaw channels and you have a working autopilot with no need to compute (or even know) the center of mass or to incorporate the moment of inertia.