You missed a window and need to log in to bethesda.net
During installation you should have noted a window popping up. As someone else pointed out. You did not? Me neither, but Discover Software Center popped up suggesting to install Mozilla Firefox. Which is odd, because Firefox was already installed.
Install Firefox as suggested and you will see the login prompt from Epic to bethesda.net on your Steam Deck. I guess this has something to do with Flatpak, which is how I installed Heroic. It seems the game wasn't able to find the browser through all the sandboxing (Wine, Flatpak).
When you have done that you should have no issues with that part anymore.
-skipmovies as a start up parameter, as suggested on ProtonDB. If you absolutely want to avoid creating a bethesda.net account you have too look a bit deeper into the problem I guess or wait about 3 minutes for the program to give up. Startup parameters don't seem to be documented very well or I'm looking in the wrong direction here.
Adding Doom 64 and other games from Heroic to Steam
Install, setup and run https://github.com/redromnon/HeroicBashLauncher/
Saves you time adding them all by hand and picking artwork.
Steam controller layout
WASD and mouse (touchpad) layout worked best for me on Stream Deck when starting Doom 64 from Steam. Using the right stick to look around resulted in terrible stutter and I don't have that much time to investigate further why.
Other games with similar issues
- Runbow - Epic Games Store or which ever store the game is downloaded from requests some permission, but the mechanism to bring up the browser or a webview does not work reliably, like on Steam Deck with Heroic