Let's restrict attention to the case where the components of $S - \Lambda$ are all discs (with three or more cusps). Following Thurston, we choose singular foliations of these discs, with the foliations being horocyclical in the cusps of the disks. These foliations-of-discs fit together to give a foliation $F$ on $S$. 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 quotient $S /{\sim}$ of $S$ is again a surface and is in fact homeomorphic to $S$. Also, the lamination $\Lambda$ descends to give a foliation $\Lambda/{\sim}$ in $S /{\sim}$. The spaces of transverse measures on $\Lambda$ and on $\Lambda/{\sim}$ are naturally isomorphic.

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. I would say that the application to counting ergodic transverse measures on laminations is "well-known to the experts".