  We investigate how vector and isovector interactions can be determined within
the density regime of neutron stars, while fulfilling nuclear and astrophysics
constrains. We make use of the Chiral Mean Field (CMF) model, a SU(3) nonlinear
realization of the sigma model within the mean-field approximation, for the
first time within a Bayesian analysis framework. We show that neutron-matter
$\chi$EFT constraints at low density are only satisfied if the vector-isovector
mixed interaction term is included, e.g., a $\omega^2\rho^2$ term. We also show
the behavior of the model with respect to the conformal limit. We demonstrate
that the CMF model is able to predict a value for the parameter $d_c$ related
to the trace anomaly and its derivative takes values below 0.2 above four times
saturation density within a hadronic model that does not include a phase
transition to deconfined matter. We compare these effects with results from
other (non-chiral) Relativistic Mean Field models to assess how different
approaches to incorporating the same physical constraints affect predictions of
neutron-star properties and dense matter equations of state. We also include
data from the gravitation wave event GW230529 detected by the LIGO-Virgo-Kagra
collaboration and the most recent radius measurement of PSR J0437-4715 from the
NASA NICER mission. Our analysis reveals that this new NICER measurement leads
to an average reduction of approximately $\sim 0.15$ km radius in the posterior
of the neutron-star mass-radius relationship.
