To rephrase your question (when m \leq d and the linear terms L_i are linearly independent): You are asking for the equisingular deformations of a normal crossing singularity. Of course, as you have observed, not all deformations of a normal crossing singularity need be normal crossings.

One comment worth making: I'm seeking a bound like $H^{100d}$ i.e. exponential in the height, with exponent linear in $d$.

The example I was thinking about was not simplicial, but is very close: The number of generators of any maximal cone exceeds the dimension by exactly 1.

Thanks for the question, I only wanted PD rationally.

I mean, I wouldn't say no to a reference which proves this explicitly. But I am asking for a stronger statement: Are the virtual fundamental cycles represented by a flat family (when the base is a curve)?