Here's a rough guide for Heroic that I just tested. Had a few free minutes so took a shot. Seems to work, but ymmv. I am currently running the flatpak version of Heroic.
Download the exe for EA App
In Heroic, download a current version of proton:
Click Add Game in the library view of Heroic
Here I've called the game "EA App". Once entered, Heroic automatically finds "box art" and enters it:
Select your wine version (I recommend the GE version) and executable location (the file we downloaded at the start).
Select Finish, then click Play on the newly created entry in the Heroic library.
EA App should pop up:
Go through install
Once this login screen opens, CLOSE the program (stop it in Heroic if once you close it, it still says it's playing)
Now we want to change the file path for this to where it just installed itself.
Click the entry to view more details
Top right of that window, should be three verticle dots. Click those, then "Edit App/Game"
Should bring you here:
For executable, your path should be something like:
/home/YourNameHere/Games/Heroic/Prefixes/default/EA App/drive_c/Program Files/Electronic Arts/EA Desktop/EA Desktop/EALauncher.exe
Once done, click Finish
Run App again, sign in as you would normally do
Download and test whatever games you want.
Currently everything on my version appears to be functioning correctly. I was able to download, install and run Dungeon Keeper (every other game was over 80 gigs, I'm not waiting on that right now lol).
If I get more free time at some point, I could do more testing. I don't own Origins on EA App, just on steam. I own a newer Dragon Age game (apparently) that I could test, but like I said, I'm not waiting forever for this brief test.