So when physicists report mass or length change as an object is speed up [down?] towards the speed of light [unity] what are they measuring?

Conventional scientists seldom include the observer principle, and work from the assumption that "what you see is what you get." That is seldom the case in Nature. For example, look at an iceberg floating in the ocean. You see a chunk of ice sticking up--what you don't see is the humongous mass, some 90% of the total volume, below the surface in the water.
Now equate that to the RS concept of the "time region," where atomic rotation takes place. It is just like an iceberg--the tip is "equivalent space," the projection of the atomic rotation that is sticking above the surface, where we can see it. But the rotation, itself, is under the surface in the depths of time, where it is unobservable by someone standing on the deck of a boat. The surface of the water is the unit speed boundary that separates motion in space (above) from motion in time (below). Larson accounts for both of these aspects when dealing with the Lorentz factors.
Let's look at the "shrinking size" situation, when an object approaches the speed of light. Common sense tells you that things don't change size depending on their speed, unless they are smashing into something so hard that the pressure squishes it. Well, in the vacuum of space, that is not a concern, so that's not a possibility.
In the RS, everything is motion, so "distance" is actually "speed," s/t, where the time component is ignored because it is unity (s/1 = s, so ignore it). If we want to make that speed go faster, it has to be
accelerated. And what does acceleration do? It adds
time to speed, s/t / t = s/t
2, putting time back into the equation. Conventional science does not recognize "time" as a part of distance because they ignore it, so they have to do something to adjust for that addition. Well... "more time" = "less space," so they shrink the distance to account for the extra time, and when the new speed is reached, they just keep the shrunken distance component.
The situation with mass increasing to infinity is the same thing, but in higher dimensions and inverted (speed is s/t, mass is t
3/s
3). Mass (for which I use Gustave LeBon's theory, which matches the RS and is better defined) is defined by angular momentum (t
2/s
2) divided by speed (s/t). The momentum is the atomic rotating system, which MUST remain fixed, or the atom will change atomic number when accelerated--and it doesn't. That means the speed component is the variable.
Remember that in the RS, things are measured from unity (speed of light), down, not from zero, up. Most matter is sitting at a fraction of the speed of light, so when a chunk of palladium is sitting on a physicist's desk, it is actually moving at nearly the speed of light, when measured FROM the speed of light. But so is the desk, and everything else, so it appears to be still. When it is thrown in a particle accelerator and sped up from zero (0+x) it is actually being slowed down (1-x). As the velocity approaches 1, the speed of light, the speed in the denominator of "mass" approaches zero. And the smaller the magnitude of the denominator, the larger the mass will appear. Even though the momentum never changed, and palladium remained palladium.
That's the beauty of the Reciprocal System--it takes into account what is in the air, on the surface and under the water (material sector, photons/waves, cosmic sector). Conventional theory deals only with the air and surface, aether theory deals only with the surface and "under the water." That is why they seem so incompatible, two sides of the same, reciprocal coin.