  Planetary debris is observed in the atmospheres of over 1,000 white dwarfs,
and two white dwarfs are now observed to contain orbiting minor planets.
Exoasteroids and planetary core fragments achieve orbits close to the white
dwarf through scattering with major planets. However, the architectures that
allow for this scattering to take place are time-consuming to explore with
$N$-body simulations lasting $\sim 10^{10}$ yr; these long-running simulations
restrict the amount of phase space that can be investigated. Here we use planar
and three-dimensional (spatial) elliptic periodic orbits, as well as chaotic
indicators through dynamical stability maps, as quick scale-free analytic
alternatives to $N$-body simulations in order to locate and predict instability
in white dwarf planetary systems that consist of one major and one minor planet
on very long timescales. We then classify the instability according to ejection
versus collisional events. We generalized our previous work by allowing
eccentricity and inclination of the periodic orbits to increase, thereby adding
more realism but also significantly more degrees of freedom to our
architectures. We also carried out a suite of computationally expensive 10 Gyr
$N$-body simulations to provide comparisons with chaotic indicators in a
limited region of phase space. We compute dynamical stability maps that are
specific to white dwarf planetary systems and that can be used as tools in
future studies to quickly estimate pollution prospects and timescales for
one-planet architectures. We find that these maps also agree well with the
outcomes of our $N$-body simulations. As observations of metal-polluted white
dwarfs mount exponentially, particularly in the era of Gaia, tools such as
periodic orbits can help infer dynamical histories for ensembles of systems.
