  We present further applications of the formalism introduced by the authors in
arXiv:2308.10949, which allows embedding of a broad class of generalized LTB
models into effective spherically symmetric spacetimes. We focus on regular
black hole models, where a broad class of models can be considered, including
for example LQG-inspired models as well as the model with a regular center,
e.g. of Bardeen and Hayward. For a certain class of regular black hole models,
we can formulate a Birkhoff-like theorem in LTB coordinates. We further show
that depending on the properties of the polymerization functions characterizing
such regular black hole models in this formalism, the uniqueness of the
effective spherically symmetric vacuum solutions might not be given in general
in Schwarzschild-like coordinates. Furthermore, we introduce a reconstruction
algorithm that allows for a subclass of this models to construct from a given
metric in Schwarzschild-like coordinates the corresponding effective
spherically symmetric model, its dynamics as an 1+1-dimensional field theory as
well as a corresponding covariant Lagrangian of extended mimetic gravity in
four dimensions. Such a reconstruction allows us to obtain Lagrangians of
extended mimetic gravity models for black holes with a regular center, e.g. the
Bardeen and Hayward metric as well as for effective LQG inspired models.
Moreover, the reconstruction enables us to extend regular black hole models to
general inhomogeneous dust collapse models. For the latter, within this
formalism, we can investigate and look at the physical properties of the models
such as the existence of weak shell-crossing singularities from a novel
perspective.
