The path $\ell$ is not well defined.  That is, there is no best path of pleated surfaces around a mapping torus.  Thus the answer to your question is "no": 
there is no upper bound on the Hausdorff distance between $\ell$ and $\sigma$, purely in terms of the topology of $S$. 

To prove this, we resort to a standard example.  Suppose that $S$ is the genus two surface.  Suppose $X$ and $Y$ are disjoint subsurfaces of $S$, both homeomorphic to the once-holed torus.  Let $\gamma = \partial X = \partial Y$.  

Let $f$ be a pA map on $X$, let $g$ be a pA map on $Y$, and let $h$ be a pA map on $S$.  Then for all sufficiently large $n$ the map $F_n = (f g)^n h$ is a pA map.   Let $M_n$ be the mapping torus for $F_n$.  Then the curve $\delta = \gamma \times \{1/2\} \subset M_n$ is very short in the hyperbolic metric in $M_n$.  Consider the following two paths $\ell$ and $\ell'$ of pleated surfaces.  The path $\ell$ first "moves through" $X$ and then moves through $Y$.  The path $\ell'$ moves through $X$ and $Y$ in the opposite order.  The paths $\ell$ and $\ell'$ are far apart in Teichmüller space, so at least one of them is far away from $\sigma$, as measured in Hausdorff distance.  (The exact same construction applies to the WP metric as well.)

Basically, the thin part of Teichmüller space contains large product-like regions.  Teichmüller geodesics "know" how to go through such regions.  Pleating paths, which are very far from being unique, do not have such knowledge.