The black hole itself would not be exploding, but it might contain an event similar (but not identical) to an explosion, that would function locally as a big bang.
These processes are described in Nikodem Poplawski's torsion-based model, elaborated in numerous papers written between 2010 and 2020, whose preprints are available free on Cornell University's "Arxiv" website, at https://arxiv.org/a/poplawski_n_1.html .
His model's based on Einstein-Cartan Theory, worked out through conversations between Einstein and the mathematician Elie Cartan that occurred after the discovery of particulate spin, and were completed in 1929. Avoiding that idealization of fermions as "pointlike" which facilitates their analysis within our own inertial frame, it differs from 1915's General Relativity by allowing for fermions to have a tiny spatial extent, which remains greater than the Planck length.
In Poplawski's model, a large rotating star, whose exhaustion of its nuclear fuel has left it without radiation pressure adequate to resist its gravitational collapse, forms a black hole, whose event horizon propagates outward from its center, with tidal effects separating the particles of virtual pairs by the Compton wavelength for more than the Compton time, thereby materializing them from the gravitational potential.
Upon contact with the stellar fermions (which would be vastly larger), the inward trajectories of many of the newly-materialized particles are reversed and greatly accelerated outward, forming a new "local universe" whose shape Poplawski has compared to the skin of a basketball. (Through Feynman's interpretation of Maupertuis' Principle of Stationary Action, from which General Relativity can reportedly be derived, such particulate interactions can be generalized in terms of spatial expansion.) After its formation, the new LU would continue expanding inertially for an indefinitely long time, but would, nevertheless, remain causally-separated from the larger LU that had contained its "parenting" star.
The initial expansion of that LU would be asymptotically exponential, so the local universes formed through Poplawski's model would be formed by a version of cosmic inflation, without any need either for the hypothetical "inflaton" particle that would be needed in those earlier versions of inflation that depend upon a field of them, or for that singularity of infinite density which characterizes the original "Big Bang" models.
Because each of the LU's would, itself, eventually contain rotating stars apt to collapse gravitationally, ad infinitum, his model may, in the resolution of disparities between that "block universe" view of time which characterizes Einstein's relativity and the quantum physics relation of time to phenomena emerging from quantum entanglement, also contain some potential for resolving the fact that "vacuum energy" is hypothesized to have a value exponentially larger than that value which can be verified by observation.
As our own local universe would've originated within one or another of the black holes in a "parenting" LU, there would be some preferred direction of motion corresponding to the direction of rotation of the axis of whichever of that LU's stars had collapsed to form our own LU, leaving some possibility for that observational verification required of scientific hypotheses. Results of the attempts at such verification that have been obtained to date remain inconclusive.
Regarding the part of the OP's question concerning the possible visibility of the quasi-explosive "bounce" to observers in the larger and older "parenting" LU outside the black hole, its larger scale and its own continuing expansion would suffice to keep the particulate interactions of that bounce permanently invisible to them, given the fact that the inertial frame of all objects remaining in the parenting LU would differ greatly from that of any of the objects involved in the bounce itself.
The "black hole information paradox" would be resolved by the transfer of information (i.e., heat) inward between "parenting" and "offspring" universes, rather than through the faint (and perhaps unverifiable) outward radiation hypothesized by Hawking. The transmitted information might, once deciphered by civilizations sufficiently advanced, even result in artifice aimed at an intermittent self-perpetuation on sequentially-decreasing scales, by such means as the addition of mass to stars about to collapse into neutron stars, which could provoke their collapse into black holes instead.
The compatibility of Poplawski's model with the Cosmic Microwave Background radiation was detailed in a 2015 paper by Desai, titled "Non-parametric reconstruction of an inflaton potential" and freely available at https://arxiv.org/abs/1510.08834 .
In that "block universe" view of time which characterizes Einsteinian relativity, the actual point at which magnification energy otherwise adequate to reveal whatever internal details a local universe might (or might not) have would (through mass/energy equivalence) also cause its collapse into a hole blacker than that point's surface: So, the unique objects chosen for representation only by a point might mark the limits of knowledge, but not of extant reality. The interesting thing is that the same curiosity motivating the assembly of any such magnification equipment would have also "illuminated" (if only by an absence of reflection) whatever intermediate black holes (each surrounding an LU) would've lain between the civilization assembling it and the object of its curiosity, which, if such illuminations would reveal a pattern, might open the possibility of a multiverse pulling itself into existence by its own bootstraps, and those of its occupants. As has been pointed out by the late John Barrow of Cambridge, miniaturization would provide one route to survival during an approach to thermal equilibrium that might be kept asymptotic.
I've been interested in Poplawski's model mainly because of a nearly life-long interest in Olbers' Paradox, which its "Black Hole Genesis" resolves very simply. Although that model's reliance on ECT may appear threatening to many physicists who have invested much time obtaining a really admirable mastery of General Relativity (Einstein-Cartan Theory's older forebear), Black Hole Genesis also provides the plainest answer to the OP's question about the lack of visible quasi-explosive effects in the context of a multiverse, and has versions based only on 1915's GR, as discussed by Brandenberger at https://arxiv.org/pdf/hep-th/0103019.pdf, and in his paper's references.