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You can find discussion of this method in the context of particle methods for advection PDEs here. Some of the results at the end of this paper will illuminate the problem. Essentially, as $\epsilon\to0$, your solution will converge in the integral sense, but pointwise estimation of this function is not so easy. If $\epsilon$ is too large, then the solution ...
After thinking about this some more, I can answer this one myself! I don't think the complex plane makes the log-sum-exp trick appreciably different, at least in Cartesian coordinates. In particular, if $z=u+iv$ then $e^z=e^{u+iv}=e^u (\cos v + i\sin v).$ Notice the $v$ part has magnitude 1 by construction, so overflow or underflow is principally caused ...