Kleene Algebra for star-free regular expressions
You might be interested in bounded synchronization delay expressions.
See [1] for details on these expressions.
To sum up, they are equivalent to star-free expressions, but instead of using complement, they restrict the use of the Kleene star to certain languages: the prefix codes with bounded synchronization delay.

This way, you can have your characterization: $L(r_1)=L(r_2)$ iff we can prove $r_1=r_2$ using the axioms of Kleene Algebra, the only difference being that $r_1$ and $r_2$ are more constrained on the use of Kleene star, and only represent star-free languages.

On the other hand, axiomatization of regular expressions with complement has been done in [2], so after careful verification, you could possibly use this to axiomatize star-free expressions directly.

[1] Diekert, V.; Kufleitner, M., Bounded synchronization delay in omega-rational expressions, Hirsch, Edward A. (ed.) et al., Computer science – theory and applications. 7th international computer science symposium in Russia, CSR 2012.

[2] 
Salomaa, A.; Tixier, V., Two complete axiom systems for the extended language of regular expressions, IEEE Trans. Comput. 17, 700-701 (1968). ZBL0174.29001.