Here’s my take on this, as it’s a situation with a couple of different angles.
Directly recovering a deleted message through Discord itself is generally not possible. When you delete a DM, you’re telling their servers to remove it from your conversation view, so there’s no “trash” folder to check on either your iPhone or PC. Most dedicated “recovery” apps struggle with this because the data isn’t just sitting on your device.
On the other hand, you do have a few practical options. The most straightforward solution is to simply ask the other person in the DM for the gaming details; the messages are only deleted for you, not for them. Looking forward, some monitoring applications, if they were already installed on your PC or phone, may have logged the conversation before it was deleted. This is often the most reliable way to have a backup of communications.