But the axioms of elementary topoi are already incomparably more complicated than the axioms for set theory presented here.
What on earth does Friedman mean by complicated? As Peter Freyd pointed out a long time ago, the axioms for an elementary topos are essentially algebraic -- that is, they live at a very low level of logical complexity. The very first axiom in anyone's (including Friedman's) axiomatization of set theory is the axiom of extensionality, which is not expressible even in coherent logic (at least, not unless you take not-membership as a primitive predicate, on the same level as membership). Unless Friedman can put forward an objective measure of complexity (as opposed to "unfamiliarity to H. Friedman") which justifies the above quote, then his challenge is not worth considering. Peter Johnstone