I don't think the information-theoretic compression-based view is at all incompatible with Wittgenstein in Philosophical Investigations, although it's absolutely against the view in the Tractatus. In the Tractatus, something is meaningful when it is isomorphic to some structured symbolic representation. Information theory is based on a rejection of that kind of view (from Shannon 1948):

"The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages."

In other words, information has nothing to do with isomorphism to a structured representation with definite relationships with the world, nothing to do with symbolic representations or any other kind of "representations". It's simply the ability to predict one thing from another thing. Shannon calls these things "messages" but mathematically the logic applies to any set: for example we can analyze an agent's policy information-theoretically, in terms of how much information about the agent's input state is contained in its output actions. I think this kind of view is highly compatible with later Wittgenstein, for whom "meaning" (if it's anything) is an abstraction of how language is used to coordinate joint action---that is, situations where people are trying to predict and control each others' behavior.

To address your examples,

> We don't call someone intelligent because we can point to some lossless compression algorithm implemented by his neurons, we call him intelligent when he produces the behaviors we desire in a given situation.

Suppose we give someone input A and he outputs desired behavior f(A), and then we give him input B and he outputs desired behavior f(B). If we give him input C will he output the desired f(C)? Only if he has learned to predict what f(C) should be as a function of C and the training data, that is, if he has learned to predict what we consider the "desired" behavior to be. In that sense, intelligence here requires prediction, which is exactly the same thing as compression.

> It seems fair to state that people with tangible skills requiring the use of the body posses some kind of intelligence, it seems less accurate to try and describe this form of intelligence as lossless compression.

Again, the agent's problem here is to select the right motor action given sensory input. Suppose you've seen a tennis ball coming in with velocity vector x and you know how to hit it to win the game, and you've seen a tennis ball coming in with vector y and you know how to hit it to win the game. What will you do with vector z? If your policy is good, it will put high probability on only actions that hit the ball in a way that makes you win. And if your policy is putting high probability on that action, that's mathematically the same as saying it provides an efficient compression of that action (high probability = small number of bits in an encoding).

This is very abstract but once you adopt this view it gives you a lot of very useful conceptual tools. For example, you can talk about the "channel capacity" of a policy in terms of how many bits of information it can "transmit" from input states to output actions, and this channel capacity turns out to be a very intuitive measure of the complexity of the policy that you can use to analyze human behavior (one example, [1]).

[1] https://gershmanlab.com/pubs/GershmanLai21.pdf