There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at the nlab page on cohomology

I would like to understand if there's any non-contrived way that Floer-theoretic invariants fit into this picture.

I'm far from an expert. I'm aware of a number of works elucidating homotopy-theoretic aspects of Floer theory: for example the construction of Cohen-Jones-Segal, and the infinity category of Lagrangian cobordisms defined by Nadler-Tanaka. Do any of these constructions provide an answer to the question above?