> I'm sure it's wrong in important ways

I'm afraid so. I'll just pick on two important ways found in one of your key sentences, and I'll stick to widely accepted results from Hawking, Unruh, Gibbons and Giddings.

> Particles at the edge of the sphere are able to leak back into the parent universe via Hawking radiation just like our galaxies at great distance slip out of our universe red-shifted faster than the speed of light.

Firstly, Hawking radiation doesn't leak anything out of the black hole itself; it's produced by interactions between the outside matter fields and the dynamical spacetime outside an evolving black hole. Secondly, you have the relevant part of cosmic horizons backwards. They also produce a form of Hawking radiation in the far future.

So, since nothing ever comes from inside the black hole's horizon even at final evaporation, or from outside the cosmic horizon, I don't see how you can recover your more cosmological ideas.

Now some technical detail:

The origin of Hawking radiation is well outside the horizon of the black hole. Quoting Unruh in <https://doi.org/10.1103/PhysRevD.78.041504> (corresponding to the preprint at <https://arxiv.org/abs/0804.1686>):

"One way of achieving a better understanding of [why black holes seem to behave by thermal objects and evaporate by emitting Hawking radiation] is to study the origin of particles in black hole evaporation, .i.e., the question of where they are created." [authors' emphasis]

The paper goes on to show that because Hawking radiation is a low-energy process, the origin of particles must be from a distance outside the horizon proportional to the radiation wavelength, and cannot originate very close to the horizon (much less from inside it). This follows from his earlier conference presentation <https://inspirehep.net/literature/775859> (pdf available at <https://pos.sissa.it/043/039/>), "Where are the particles created in Black Hole evaporation?"

Giddings makes the same point in <https://www.sciencedirect.com/science/article/pii/S037026931...> (open access, but easier to read as a preprint at <https://arxiv.org/abs/1511.08221>). From the abstract:

"Where does Hawking radiation originate? A common picture is that it arises from excitations very near or at the horizon ... However, closer investigation of both the total emission rate and the stress tensor of Hawking radiation supports the statement that its source is a near-horizon quantum region, or "atmosphere," whose radial extent is set by the horizon radius scale".

The last clause there is expanded in the text:

"... the source of Hawking radiation is a quantum region of size \Delta r ~ R outside the black hole horizon"

R there means the Schwarzschild radius, so particles originate in a fairly voluminous region with the lower edge at about a Schwarzschild radius above the black hole's horizon, i.e., at about 2R.

On Hawking radiation from cosmic horizons, Gibbons and Hawking's 1977 paper 1<https://journals.aps.org/prd/abstract/10.1103/PhysRevD.15.27...> details the production there. You can get this paper from <https://sci-hub.ru/https://doi.org/10.1103/PhysRevD.15.2738>.

"We have shown that the close connection between event horizons and thermodynamics has a wider validity than the ordinary black-hole situation in which it was first discovered. An observer in a cosmological model with a positive cosmological constant will have an event horizon whose area can be interpreted as the entropy or lack of information that the observer has about the regions of the universe that he cannot se. When the solution has settled down to a stationary state, the event horizon will have associated with it a surface gravity K which plays a role similar to the temperature in the classical first law of event horizons ... this similarity is more than an analogy: The observer will detect an isotropic background of therman radiation ... coming, apparently, from the event horizon".

The 21st century updates from Unruh and Giddings applies to this 1977 paper by Gibbons and Hawking: "from the event horizon" -> "from a region closer to the observer than the horizon".

The above assumes General Relativity is a good physical model at the length scales in question. If you don't assume that, then you might be able to ignore the results above if black holes and expanding space are substantially different from their standard descriptions in General Relativity. How different? Would the recent data from the Event Horizon Telescope still be produced? Would the waveforms from binary black hole and black hole-neutron star mergers found by LIGO and Virgo still be produced?
Would the cosmic microwave background have the same temperature and power spectrum?