Answering my own question.
It is known that the Lindelöf sum of the Mercator series, whose terms are $-(1-z)^n/n$ here, converges for all $z≠-1$ with magnitude 1, as they lie strictly on the interior of the Mittag-Leffler star(w.l.o.g, as the case when $z=1$ is trivial). Therefore this converges for all orthogonal matrices $Z$ (with no eigenvalue being exactly $-1$) by the spectral theorem. Setting $Z=exp(K)exp(L)$, one can simply Lindelöf sum the BCH-D series.
This holds if $exp(K)exp(L)$ does not have any eigenvalues equal to $-1$. If it does, one can always use the (trivial) formula $ln(exp(K)exp(L)) = ln(exp(K)exp(L+2πiaI))$$-2πiaI$, $ln$ being in any suitable branch and choosing real $a$ such that $exp(K)exp(L)exp(2πia)$ does not have any eigenvalues equal to $-1$(such an $a$ trivially exists), and use the fact that the BCH-D series of $ln(e^Xe^Y)$ is equal to $X+Y$ plus nested commutators. Since commutators are invariant w.r.t. translation re: each component by $a*I$, one returns to the Lindelöf sum of the original series, even if $exp(K)exp(L)$ has any eigenvalues equal to $-1$. Q.E.D.