The complexity of the dominating set problem in specific subclasses of chordal graphs
Too bad you have been waiting so long without getting any answer.
I don't know for the classes you asked for, but I know some related graph classes and new techniques you can try.

First I will mention that Steven Chaplick has done work on related graph classes, he finished his thesis earlier this year, you might find his research interesting.

I know some results in this direction follow from my own work
Graph Classes with Structured Neighbourhoods and Algorithmic Applications
This gives a general technique for solving various problems including DSP in certain graph classes. 
We do this by introducing new graph decompositions (see my thesis).

For instance if we are give an intersection model of UP where the tree has max degree d and the paths have max length s we can solve dominating set in $(d-1)^{3(s-1)}poly(n)$.

Similar if we have a graph given with an intersection model consisting of $0$-bend paths in a $k \times n$ grid (for constant k).

The same technique might work for CEPT arising form a host tree of maximum degree 3, but I need some more time to understand this class.
If you have a link to some characterizations of this class that would help.