= Solution
The scalar kinetic operator is proportional in momentum space to $k^2(\widetilde c k^2-m^2)$. Its inverse has the partial-fraction decomposition
$$
\frac{i}{k^2(\widetilde c k^2-m^2)}
=-\frac{i}{m^2}\left(
\frac1{k^2}-\frac{\widetilde c}{\widetilde c k^2-m^2}
\right),
$$
up to the overall convention-dependent factor allowed in the question. Thus $\widetilde c=\widetilde a+\widetilde b$. Under the physical relation $\widetilde b=-\widetilde a$, one has $\widetilde c=0$, so the extra massive pole and its ghost residue disappear.
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