Electric Bass with RJ45 Digital Output
I'd just like to understand if this is a rabbit hole or the elusive unobtainium.

A little bit of both. Ethernet is really NOT a great choice for real time audio. You would need a fairly beefy processor that can handle a network stack (TCP or UDP). Getting the latency down to something that's playable is extremely challenging.
Even if you get it on Ethernet, where would it go ? There some multi-channel network formats but these tend to be in the PRO space  and expensive.
If you just care about the connector, you can certainly run I2S or TDM through an RJ45 + Ethernet cable. That works quite well (and I've actually managed to incorporate this into a commercial product). Again, you would need some type of receiver that's matched to your transmitter.