Fitting a regular grammar to strings from a PCFG: how big does it get?
The question, if understood in the simplest naive way, might be uninteresting.

Here is a simple example. Let $G$ be the regular language $\{a^n\mid n\ge0\}$ over the alphabet $\{a\}$. 

Consider the strings $\epsilon, a, a^2, \cdots, a^n$ in $G$. What are the regular languages that contains those strings?


The minimal such language, i.e., which contains no other strings, will need $n+1$ generation rules $S\to a^i$ for $0\le i\le n$.
The language with the least generation rules is $G$ itself, which has two generations rules $S\to \epsilon$ and $S\to aS$.




However, once we start to twiddle with the ways how to approximate context-free grammar by regular grammar, there are tons of research. 

Here is a related question that links to many related stuff, Is there a known method for constructing a grammar given a finite set of finite strings?.

This paper considers approximating CFG from above by a regular
grammar.

You can browse the google search result for approximate context-free grammar by regular grammar or the google scholar.