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After some reading and contemplation I realized I was going about this totally the wrong way. Trying to get Docker to behave like Proxmox was the big mistake here. There appears there are ways to expose a Docker network so that external computers can access the containers by hostname by proxy-ing DNS through the Docker network. That doesn't solve the DHCP issue though.
 
The real solution was to not do that at all. I ended up creating a custom bridge network in Docker and put my containers on that network. This allows the containers to reach each other by their container name (as long as they are on the same network). This makes setting up services that reference each other (like the various *arrs) much easier. To access my containers on my LAN/WAN I used reverse proxies for most services and for the few services I wanted to directly access locally (such as streaming services) I forwarded their ports to the host and access them that way.
 
Coming from Proxmox put me in the wrong headspace and after wrapping my head around unRAID/Docker it made things way clearer and ultimately way easier.