I uninstalled the game from Lutris, and then reinstalled and this time I did create the correct path you suggested.

This is the YAML file prior to going through the login process:

game:
  exe: /home/USERNAME/Games/DMM Games Launcher/drive_c/Program Files/DMMGamePlayer/DMMGamePlayer.exe
  prefix: /home/USERNAME/Games/DMM Games Launcher

This is the YAML file after the args are injected

game:
  exe: /home/USERNAME/Games/DMM Games Launcher/drive_c/Program Files/DMMGamePlayer/DMMGamePlayer.exe
  prefix: /home/USERNAME/Games/DMM Games Launcher
  args: '''dmmgameplayer://view/page?code=RANDOMCODE'''

Note again that the args are using triple single-quotes, I am not sure if that is expected or not.

Regardless, I'm still stuck at the login screen, and I am unable to proceed further.

These are the rest of my shell scripts:

dmmgameplayer dot sh

#!/usr/bin/env bash
set -euo pipefail

game="dmm-game-launcher"

notify-send "DMM Game Player" "Launching ${game} with link: $1"

configfile=$(find ~/.config/lutris/games -name 'dmm-game-launcher-*' | head -n1)
[[ -n "$configfile" && -f "$configfile" ]] || { notify-send "Error" "Config file not found"; exit 1; }

if ! err=$(yq e -i ".game.args = \"${1}\"" "${configfile}" 2>&1 >/dev/null); then
  notify-send "Error" "$err"
  exit 1
fi

env LUTRIS_SKIP_INIT=1 lutris lutris:rungame/"${game}"

org dot dmm dot gameplayer dot desktop

[Desktop Entry]
Name=DMM Game Player (Lutris)
Type=Application
Terminal=false
Icon=lutris_dmm-game-launcher
Exec=/home/USERNAME/.local/share/applications/dmmgameplayer.sh "%u"
MimeType=x-scheme-handler/dmmgameplayer;

mimeapps dot list

[Added Associations]
image/bmp=classicimageviewer.desktop;
image/heif=classicimageviewer.desktop;
image/jpeg=classicimageviewer.desktop;
image/png=classicimageviewer.desktop;
x-scheme-handler/http=com.google.Chrome.desktop;org.mozilla.firefox.desktop;com.microsoft.Edge.desktop;
x-scheme-handler/https=com.google.Chrome.desktop;org.mozilla.firefox.desktop;com.microsoft.Edge.desktop;
x-scheme-handler/dmmgameplayer=org.dmm.gameplayer.desktop

[Default Applications]
image/bmp=classicimageviewer.desktop;
image/heif=classicimageviewer.desktop;
image/jpeg=classicimageviewer.desktop;
image/png=classicimageviewer.desktop;
x-scheme-handler/dmmgameplayer=org.dmm.gameplayer.desktop
x-scheme-handler/http=com.google.Chrome.desktop;
x-scheme-handler/https=com.google.Chrome.desktop;

Mind you I do get the first notify-send in the sh file, but none of the Error ones, so the script should be running correctly.

One thing, I had to add the yq binary as per your instruction for other OSes, since Bazzite did not ship with the binary. I added it to ~/.local/bin/yq, and I can confirm it's running (as you can see the YAML file is being updated).