Does the idea that everything is made of causes and effects meaningfully constrain experience?

A causally disconnected event would not affect our experience, so no, it does not constrain our experience. But it wasn't meant to: it was meant to constrain our maps, not our experience. To stop us from believing meaningless statements.

Can you coherently say how reality might look, if our universe did not have the kind of structure that appears in a causal model?

Every causally separated group of events would essentially be its own reality.