I believe the two constructions should coincide. Here's why.

There is an interesting and very close variant of the Dirichlet domain construction you're describing. Take a singular flat metric on the fiber surface $F$, in which the stable foliation is vertical, and the unstable foliation horizontal. Now, consider a maximal rectangle $R \subset F$, with sides along the leaves of the foliations, and whose interior is disjoint from the singularities. By maximality, this rectangle must have a singular point along each of its 4 sides, which defines a quadrilateral in $F$. 

Thickening the rectangle gives a tetrahedron. Let's say, by convention, that the diagonal connecting horizontal sides is above the diagonal connecting vertical sides (this matches the verticality of the stable foliation). It's not hard to check that these tetrahedra glue up to give a natural, combinatorially canonical triangulation of $F \times [0,1]$, which is invariant under the monodromy. 

Guéritaud has proved that the above construction produces the same canonical triangulation as Agol's train track construction. I don't believe his proof is written up, but an announcement (including a summary of the construction) appears in this Oberwolfach report:

http://www.mfo.de/occasion/1218/www_view

Now, it remains to ask: does Guéritaud's construction produce the the same result as your Dirichlet domain construction? It seems to me that they should coincide. For Guéritaud, triangles in $F$ come from rectangles in the singular flat metric with two singular points on adjacent sides and one singular point in the opposite corner. It seems likely that after rescaling in the horizontal or vertical direction, those 3 points would lie on a circle whose interior is free of other singular points, i.e. those 3 points define a cell of the Dirichlet domain. Then, one would need to check that as one moves along the Teichmuller line, i.e. rescales in the horizontal and vertical direction, the inscribed circles with empty interior transform in the right way.

The above paragraph is not a proof, but it seems very plausible that one could construct a proof along these lines.