Does the idea that everything is made of causes and effects meaningfully constrain experience? Can you coherently say how reality might look, if our universe did not have the kind of structure that appears in a causal model?

No. I literally assign prior probability zero to the statement that the universe is not made out of causes and effects, because there is causal structure in Turing machines and in all Turing-complete models of computation which could make up the Solomonoff prior. Causal structure is a very broad thing - it's just a sparse graph of interacting entities with a lattice ordering.

I can imagine a universe in which the local ordering I observe doesn't go as far forward or back as I thought, and the true everything-is-causes-and-effects structure is pushed one layer back to something completely hidden from me. I can imagine a universe in which I've falsely inferred an ordering which isn't there, and getting confused by cycles in a graph that I thought was causal. But a universe with no causality at the lowest layer - I think causality is inherent in too many things, and that after subtracting those things there's not enough option space left to make a universe out of.