This will not simulate non-classical correlations then. You either need to have superdeterminism, retrocausality (i.e. actual information carrying physical interaction moving back in time), nonlocality (actual interaction at a distance), rejection of Reichenbach's principle (specifically the no decorrelating explanation part), rejection of single outcomes or reject the concept of the fundamental dynamics being mathematical (i.e. anti scientific realism).Again chaotic dynamics doesn't have to get around any of that. I will try to answer each point, by stating what I am assuming/hypothesizing:
1) Analyticity/holomorphicity making single outcomes a necessity per the conventional uniqueness and existence arguments.
2) No assumption of superdeterminism.
3) No assumption of retrocausality of matter or information.
4 & 5) The physical common cause is the specific spacetime pathway connecting some EPR pair
6) All matter and information can only follow timelike and lightlike curves....
Notice that I am not saying that the above arguments are correct per se, but merely a logically valid possibility which is mathematically speaking completely conceivable and possibly even already directly constructable using existing mathematics.
It doesn't matter if the dynamics is chaotic, dissipative and more nonlinear than anything ever conceived, unless one of these things is true Bell's theorem guarantees it will fail to replicate non-classical correlations.