I found the cause for this.
I've had this issue with PS4 and PS5 controllers on PC and finally identified the problem.
Epic and Steam use different IDs for the same device.
They do use the same folder in the windows user profile and access the same files, but in the game it only shows binding profiles that match available devices.
For example, my PS5 controller is identified as "054C0CE6" by E

in Epic, but as "28DE11FF" in Steam. Therefore, the key binding I created in the Epic version, which references the device ID 054C0CE6, will not show when I launch E

through Steam, where the game identifies the very same controller as 28DE11FF. Here an excerpt of the bindings file:
<YawAxisRaw>
<Binding Device="28DE11FF" Key="GamePad_LStickX" />
One possible solution, depending on the controller device, is to create a copy of the bindings file, open it in
Notepad and replace all references to the device ID with the ID used by the other service. A quick ctrl+h and Replace all does that in 5 seconds.
This solved it for my PS4 controller, but not for the PS5 controller. A second layer of inconvenience can be the "Key" variable. In my case, for the PS5 controller Steam and Epic also use different labels for the keys/buttons on the controller.
Example YawAxisRaw
<YawAxisRaw>
<Binding Device="054C0CE6" Key="Joy_XAxis" />
<YawAxisRaw>
<Binding Device="28DE11FF" Key="GamePad_LStickX" />
You can see in Epic the same axis of the right stick is called
Joy_XAxis, while Steam gave it the less than identical label
GamePad_LStickX. Further complicating the transition. In this case the binding profile is visible in the other service, but empty where gamepad buttons were defined.
I went on to manually set up my steam binding and make sure backup both versions for my Epic and Steam profile, but it seems an easier way to workaround these differences in the services it so simply run them both via Steam.
This has driven me crazy for many months now and I couldn't find a clue to this anywhere. So, I hope this gives some insight to those with the same problem.