Missing public exponent
The hint as stated in the question is that the missing public exponent $n_5$ was constructed "poorly". But it could be that some other public exponents $n_s$ were constructed similarly, and this could reveal what "poorly" is. I'd check if some pattern emerges from


the seven other $n_s$
the five possibly different $n'_s=n_s\bmod\Gamma(R_s)$ for $s$ such that, further, $R_s$ is of known factorization as $P_s^\,Q_s$ so that we can compute the quantities $\Gamma(R_s)=\operatorname{lcm}(P_s-1,P_s+1,Q_s-1,Q_s+1)$
the five $m_s={n_s}^{-1}\bmod\Gamma(R_s)$ for such $s$.


If nothing comes out of it, we can try to decrypt with guesses of $n_5$ and see if the plaintext's characteristics match those of the other plaintexts (e.g. range, padding, being composed in a certain alphabet/language..). Reasonable guesses could be:


the seven other other known $n_s$
the know $n'_s$ if different
the few smallest positive $n$ that are coprime with $\Gamma(R_5)$
$n=m^{-1}\bmod\Gamma(R_s)$ for the few smallest positive $m$ that are coprime with $\Gamma(R_5)$. This simplifies into raising the ciphertext to increasingly large powers $m$, and further speedups are possible for that by precomputing the ciphertext to small even powers.


I do not see that we could mount a discrete logarithm attack as considered in the end of the question, for lack of anything suggesting known plaintext.