Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$?

Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\sigma>1/2$ and $|t|\leq 5/8$ (note: much more is known for $q\leq 200000$ or so), I can give a bound of the form $$\left|\frac{L'(1,\chi)}{L(1,\chi)}\right| \leq \frac{5}{2} \log M(q) + 15.1$$ (constants not optimal) using Landau/Borel-Carathéodory, where $M(q) = \max_n |\sum_{m\leq n} \chi(m)|$, and then of course I can bound $M(q)$ using Pólya-Vinogradov (in its original form or one of its stronger, more recent variants), but I was wondering whether there was a simpler and/or more standard way. (Or, perhaps, who knows, even a closed expression I ought to know but don't.)

Thank you very much for all the very good answers - I've left comments below. Here is a remark making reference to the accepted answer (Lucia's).

Lucia says: "the constant $B(\chi)$ is a little tricky to bound". In fact, Lucia's answer, which avoids using $B(\chi)$, gives a very good bound on $|L'(1,\chi)/L(1,\chi)|$... and thus on $B(\chi)$. Let me explain the implication. Write $b(\chi)$ for the constant coefficient of the Laurent expansion of $L'(s,\chi)/L(s,\chi)$. Using the functional equation, one can easily prove that, for $q>1$, $$b(\chi) = \log \frac{2\pi}{q} + \gamma - \frac{L'(1,\overline{\chi})}{L(1,\overline{\chi})}.$$ It is immediate from Lucia's equation (1) and the Laurent expansion $\Gamma'(s)/\Gamma(s) = -1/s - \gamma + (\dotsc) s$ that $$b(\chi) = - \frac{1}{2} \log \frac{q}{\pi} + \frac{\gamma}{2} + B(\chi).$$ Hence $$B(\chi) = \frac{1}{2} \log \frac{4 \pi}{q} + \frac{\gamma}{2} - \frac{L'(1,\overline{\chi})}{L(1,\overline{\chi})}.$$

Thus, Lucia's bound implies that $B(\chi)\leq \frac{3}{2} \log q$, up to a check for small $q$ (and should give $B(\chi)\leq (1+\epsilon) \log q + c_\epsilon$ with $c_\epsilon$ explicit in general. Moreover, since $L'(1,\chi)/L(1,\chi) = o(\log q)$ in reality (conditionally on GRH), it must actually be the case that $B(\chi) = (1/2 + o(1)) \log q$.

I take these bounds on $B(\chi)$ must be known?