Let's restrict attention to the case where the components of $S - \Lambda$ are all discs (with three or more cusps).  (In particular, I assume that $\Lambda$ has no leaves which are simple closed curves.) Following Thurston, we choose singular foliations of these discs, with the foliations being horocyclic in the cusps of the disks.  These foliations-of-discs fit together to give a foliation $F$ of $S$.  Note that $F$ is transverse to $\Lambda$.  (If you arrange matters carefully, you can ensure that $F$ contains your loop $I$.)  We now collapse all leaf segments of $F - \Lambda$.  It is a (difficult) exercise to check that the resulting quotient $S /{\sim}$ is again a surface and is in fact homeomorphic to $S$.  Also, the lamination $\Lambda$ descends to give a foliation $\Lambda/{\sim}$ of $S /{\sim}$.  The spaces of transverse measures on $\Lambda$ and on $\Lambda/{\sim}$ are naturally isomorphic.

The collapsing argument can more-or-less be found in the (very readable) book [Automorphisms of surfaces after Nielsen and Thurston][1].  I would say that the application to counting ergodic transverse measures on laminations  is "well-known to the experts".


  [1]: https://www.cambridge.org/core/books/automorphisms-of-surfaces-after-nielsen-and-thurston/2AD58B246E36B971CCB92BD5B923BDB9