Thanks to everyone who has contributed to this solution, and especially to @HuSoS for all of the detail. I used the info in your post but took a different route, since my dockers all have unique IP addresses on a custom network and I wanted to use Unraid's GUI rather than the compose plugin. Here's the write-up:
Follow these steps from HuSoS's post above, replacing the internal IP address for signature_server in Step 6 with the custom address I would be assigning to the helper:
Create a new container for the helper via the GUI:
Docker > Add Container
Switch to Advanced View
Populate the following field values (icon is borrowed from Invidious docker, and optional):
Name: inv-sig-helper
Repository: quay.io/invidious/inv-sig-helper
Icon URL: https://raw.githubusercontent.com/Joshndroid/joshndroid-unraid-docker-templates/main/icons/invidious.png
Post Arguments: --tcp 0.0.0.0:12999
Network Type: [YOUR CHOSEN NETWORK | e.g.- Custom : br0]
Fixed IP address (optional): [STATIC IP ADDRESS | e.g. - 10.0.1.111]
Add another Path, Port, Variable, Label or Device
Config Type: Variable
Name: RUST_LOG
Key: RUST_LOG
Value: info
Done
Verify that the inv-sig-helper docker has successfully started (last line in the log should end with "Successfully fetched player")
Start the invidious container, verify that the logs show no errors, and enjoy!