There are various versions of stratified Morse theory out there, the granddaddy being the Goresky-Macpherson work. But for these purposes you're fine just using Morse theory on manifolds with boundary, which is a straightforward generalization of plain old Morse theory. In this setting you get Alexander duality "seen" at the chain-complex level in some sense, kind of analogous to Poincare's proof using dual cell decompositions to a triangulation, of the Morse theory/handle proof you refer to.
It's a little different though. The idea is to take the standard height function on $S^n$, and to perturb it so that it restricts to a Morse function on $M \subset S^n$, and on the boundary of a smooth tubular neighbourhood to $M$. Critical points to the height function, restricted to $M$ split into two kinds -- the kinds that contribute cells to the decomposition of the tubular neighbourhood, and the kinds that contribute to the decomposition of the complement. So a $k$-cell for $M$ corresponds to a $(n-k-1)$-cell for the complement. And so on.
I can say more later.