There's has been some work in Computational Linguistics on 'regular approximations' of context-free grammars -- converting CFGs into roughly equivalent regular grammars. A lot of this revolves the notion of a 'strongly-regular-grammar', this is a CFG that doesn't have self-embedding rules, such as A --> alpha A beta (like your rules), where alpha and beta are non-empty. Grammars with self-embedding are not regular languages, the problem being that further expansions of A by the same rule creates unbounded communication between the left and right, i.e. alpha^n A beta^N. In the other direction, Chomsky proved (the original paper is hard to find, see Harrison's book for the proof) that CFGs without self-embedding rules are regular (see this paper for a constructive proof).
The basic idea for dealing with self-embedding CFG grammars, like the ones you mention, is to convert them to strongly regular (i.e. non self-embedding) grammars -- there are efficient algorithms for doing this e.g. here, see this for review, and citations to original work) . The resulting 'strongly regular' grammar can then be converted into a finite-state automata, or perhaps some other type of regular grammar you're interested in.