Finally got this working. The issue is the container must have a 1:1 volume mapping. This was difficult because I did not want the server files filling up the Unraid docker partition. My solution was using symbolic links within the container. I'm not very experience with docker, so if you see an issue with what I have done below, please let me know. I hope this helps those who experience the same issue with the daemon. At least until the Pterodactyl team is able to change the need for 1:1 volume mapping. My guess is it has something to do with the daemon using docker to create new containers.
Solution:
A) A volume of /tmp/pterodactyl will be mapped to the Unraid host location /tmp/pterodactyl. It's important to create a symbolic link first. I tried getting around this, but I found no other solution.
SSH into Unraid as root
Make a new directory
mkdir /mnt/user/appdata/pterodactyl/daemon-tmp
Create symbolic link from /tmp/pterodactyl to the directory above
ln -s /mnt/user/appdata/pterodactyl/daemon-tmp /tmp/pterodacty
B) When setting up the template for the docker container, the following changes need to be made
Remove all default mappings from the template (config, daemon-data, tmp, etc)
Add new path mappings below (the important part is keeping it 1:1):
/mnt/user/appdata/pterodactyl/daemon : /mnt/user/appdata/pterodactyl/daemon
/mnt/user/appdata/pterodactyl/daemon-data : /mnt/user/appdata/pterodactyl/daemon-data
/mnt/user/appdata/pterodactyl/daemon-tmp : /mnt/user/appdata/pterodactyl/daemon-tmp
/tmp/pterodactyl : /tmp/pterodactyl
Accept the new template configuration and start the docker container
With the Pterodactyl-Panel, create a new node with the config path as
/mnt/user/appdata/pterodactyl/daemon-data
Write the config generated by the Pterodactyl-Panel as core.json to
/mnt/user/appdata/pterodactyl/core.json
SSH into Unraid as root again (or pickup the session from before)
Copy the core.json file into the pterodactyl-daemon container
docker cp /mnt/user/appdata/pterodactyl/core.json pterodactyl-daemon:/srv/daemon/config/core.json
Restart the Pterodactyl-Daemon container with Unraid
Execute a new shell in the container with
docker container exec -it pterodactyl-daemon sh
Copy contents of /srv/daemon to the /mnt/user/appdata/pterodactyl/daemon
cp -r /srv/daemon/* /mnt/user/appdata/pterodactyl/daemon/
Remove the default daemon folder
rm -r /svr/daemon
Create the symbolic links for daemon and daemon-data
ln -s /mnt/user/appdata/pterodactyl/daemon /srv/daemon
ln -s /mnt/user/appdata/pterodactyl/daemon-data /srv/daemon-data
Exit the container and restart the Pterodactyl-Daemon container with Unraid
Enjoy creating new servers!
Since the permissions are setup by the daemon after you create a server in the panel, I copied my custom server files into the /mnt/user/appdata/pterodactyl folder and SSH'd into Unraid to move them from there. I'm sure you can change the permissions of the folder, but I wanted to preserve the permissions as is since I won't need to edit them afterwards.
Thanks!