It's quite fitting that this conclusion comes 1 year from the start of my fanciful idea.
After the initial installation, I had to actually install my home server and NAS into the Core. I had hoped to fit the battery backup in as well but it caused some issues with casting shadows so I had to leave it on the outside. This means that now the whole Core has a battery backup so that's a pretty nice turn of events. Let me tell you, it's a lot easier to work on the wiring inside this core when it's still half open rather than working just through the hatch.
The core has the large central chamber for the electronics to fit along with a central column going through the top ring stack straight out to the top. It was my hope that the warm air from the server would funnel through this column like a chimney and vent out through the gap along the top of the unit. I was disabused of this notion after the first day of running my server. After just a few hours I was notified by my server that the temperature had exceeded acceptable levels and I was forced to leave the front hatch off to allow sufficient airflow.
This was always a possibility so I had a fallback option. I acquired a 120mm silent PC cooling fan and wired it in directly to a plug so it would be permanently on as long as the Core had power. I then mounted this at the entrance to the central column of the top stack to force the air to behave as I envisioned. This seemed to work out swimmingly. With the addition of the fan, the server temps became extremely stable and even look better than they ever did sitting out in the ambient environment of my living room.
With this last hiccup fixed, the core could be sealed and now fulfill its intended role in all its glory.
Success! It's done!
...or is it?
There are still a few minor additions I'd like to make to the core including some nicer knobs and updating the code to respond to commands from my home automation system (Home Assistant). However, I feel comfortable declaring this the successful conclusion of the original scope I had envisioned. Today's a good day.