What is the relation between universality and chaotic behavior in Elementary Cellular Automata?
Chaotic behavior does not seem to imply universality, from what I can see.  I don't see any reason to expect it would.

For instance, here's an example to challenge your intuition that the two concepts are somehow related.  Wikipedia lists the dynamic system given by the map $x \mapsto 4x (1-x)$ and $y \mapsto x + y \bmod 1$ as an example of a chaotic system.  However I doubt that this system is universal in any meaningful sense.

As David Richerby wrote, the way you prove universality is "by proving that the thing can simulate some other universal system".  I don't think "chaotic behavior" -- whatever that might mean -- is a helpful guide to determining whether a particular system is universal.