Can something similar be said for the remainder of the cumulant
generating function $\log\mathbf E e^{itX}$ with an error bound in
terms of cumulants?

**Yes and no.**
Your question is whether the union bound for Fourier transform can be somehow generalized to log Fourier transform for the class of $L^{n+1}(\mathbb{R})$ (for simplicity I do not discuss $\mathbb{R}^n$ domain case).

**"Yes part".**

The result you are asked for is basically a Tauberian-type theorem for log Fourier transform that controls the residue in terms of coefficients($\kappa_n$ as in OP). For log Laplace transform $\log\mathbf E e^{tX}$, the result is readily available in Theorem 16.1 of [Korevaar].
We write the density $p(\xi)=dP(\xi)$ and the Laplace transform
$\mathcal{L}dP(\xi)=\mathcal{L}dP(\frac{1}{x})=\int_{0^{-}}^{\infty}e^{-\frac{v}{x}}dP(x)$ according to Part IV Thm 16.1, we have
$log\mathcal{L}dP(\xi)=log\mathcal{L}dP(\frac{1}{x})\sim(\alpha-1)\frac{x^{\alpha}L(x)}{x}$ as $x\rightarrow\infty$ where $L(x)$ is the Laplace transform. For harmonic $P$, there are some results similar contained near the end (possibly title "random functions" or something like that, do not have the book at hand now) of [Paley-Wiener]. [Safarov] provides some concrete applications under certain assumptions on $P$'s (say 1.20-1.21). In this regard, "complex Tauberian" or "Fourier Tauberian" are the correct keywords to search for.

**"No part".**

Although there are some results in spirit of Tauberian theorems as mentioned above, I do not know there is a result concerning a class as general as $L^{n+1}(\mathbb{R})$.
As you already noticed there is such a potentiality of singularities that prevents you from bounding the residues, say Schwartz-class measures and their power polynomials. If you translate the bound on moment generating function $\mathbf E e^{i t X}$ onto $\log\mathbf E e^{itX}$ using the relation between moments and cumulants (see, Mobius (inverse) transform [wiki, Rota]), you will still have $\pm$ cancellation in these relations;
However, there is still arising research concerning bounding the tail behavior of a given density using cumulant expansions [Valz et.al], so I guess this is not completely a dead end.

*This is also (approximately) a question puzzled me a year ago. This answer is my exploration, and I really hope someone more knowledgeable can provide a more complete answer. Hope this helps!*

**Reference**

[Korevaar]Korevaar, Jacob. Tauberian theory: A century of developments. Vol. 329. Springer Science & Business Media, 2013.

[Paley-Wiener]Paley, Raymond Edward Alan Christopher, Norbert Wiener, and Norbert Wiener. Fourier transforms in the complex domain. Vol. 19. New York: American Mathematical Society, 1934.

[Safarov]Safarov, Yu. "Fourier Tauberian theorems and applications." Journal of Functional Analysis 185.1 (2001): 111-128.

[wiki] https://en.wikipedia.org/wiki/Cumulant

[Rota]Rota, Gian-Carlo, and Jianhong Shen. "On the combinatorics of cumulants." Journal of Combinatorial Theory, Series A 91.1-2 (2000): 283-304.

[Valz et.al]Valz, Paul D., A. Ian McLeod, and Mary E. Thompson. "Cumulant generating function and tail probability approximations for Kendall's score with tied rankings." The Annals of Statistics (1995): 144-160.

[Bazant]Martin Bazant, Lecture 5: Asymptotics with Fat Tails. https://ocw.mit.edu/courses/mathematics/18-366-random-walks-and-diffusion-fall-2006/lecture-notes/lec05.pdf