  We study the capacity of Bayesian Neural Networks (BNNs) to detect new
physics in the dark matter power spectrum. As in previous studies, the Bayesian
Cosmological Network (BaCoN) classifies spectra into one of 5 classes:
$\Lambda$CDM, $f(R)$, $w$CDM, Dvali-Gabadaze-Porrati (DGP) gravity and a
'random' class, with this work extending it to include the effects of massive
neutrinos and baryonic feedback. We further develop the treatment of
theoretical errors in BaCoN-II, investigating several approaches and
identifying the one that best allows the trained network to generalise to other
power spectrum modelling prescriptions. In particular, we compare power spectra
data produced by EuclidEmulator2, HMcode and halofit, all supplemented with the
halo model reaction to model beyond-$\Lambda$CDM physics. We investigate
BNN-classifiers trained on these sets of spectra, adding in Stage-IV survey
noise and various theoretical error models. Using our optimal theoretical error
model, our fiducial classifier achieves a total classification accuracy of
$\sim$ 95% when it is trained on EuclidEmulator2-based spectra with
modification parameters drawn from a Gaussian distribution centred around
$\Lambda$CDM ($f(R)$: $\sigma_{fR0} = 10^{-5.5}$, DGP: $\sigma_{r\mathrm{c}} =
0.173$, $w$CDM: $\sigma_{w0} = 0.097$, $\sigma_{wa}=0.32$). This strengthens
the promise of this method to glean the maximal amount of unbiased
gravitational and cosmological information from forthcoming Stage-IV galaxy
surveys.
