Ok so I got it working. Here were the steps to achieve it.
1. My project was blueprint only, which is apparently causes an issue. As suggested by Fabien, "To convert your project from blueprint only to C++, simply navigate to File > New C++ Class > World Settings and choose "Create Class". Visual Studio will open, you just need to close it and then you are good to go." However, this didn't conclude correctly and I had to rebuild UE4 from source, by opening the newly created solution (PROJECT_NAME.sln file at the root of your project) in Visual Studio.
2. This raised another issue of " Unable to expand variable in '$(DXSDK_DIR)\Lib\x64' " when trying to build the project. I found help on the UE4 Answer Hub here which makes you replace a line in the Wwise plugin source code. This made me follow a third step that I honestly think wasn't necessary to solve the issue, so you might want to jump to step 4.
3. I installed the Wwise plugin as an engine plugin following this tutorial, following only step 1 since somehow steps 2 and 3 could not be completed on my end (folders not existing, plugin not appearing in UE4 editor, things like that).
4. Open file AkAudio.Build.cs found in PROJECT_NAME/plugins/Wwise/source/AkAudio, go to line 250 or so and replace :
PublicLibraryPaths.Add("$(DXSDK_DIR)" + Path.DirectorySeparatorChar + "Lib" + Path.DirectorySeparatorChar + LibFolder);
with:
string DXSDK_DIR = Path.GetFullPath(Path.Combine(EngineDirectory, "Source/ThirdParty/Windows/DirectX"));
PublicLibraryPaths.Add(DXSDK_DIR + Path.DirectorySeparatorChar + "Lib" + Path.DirectorySeparatorChar + LibFolder);
If you followed step 3 you might want to do the same change in the AkAudio.Build.cs found in your Unreal installation directory under UnrealEngine/UE_4.20/Engine/Plugins/Wwise/Source/AkAudio. All credit go to the community of the UE4 Answer Hub as linked in step 2.
5. Rebuild your project (and optionally Unreal Engine too) and it should open normally, at least it did for me. Packaging the game now doesn't generate the error anymore and the bank is loaded correctly with my sounds playing.
Thanks to Fabien and the UE4 community I can now package my game and still use Wwise in my project =) I hope this helped you too. This seems to be only a recent issue with UE 4.20 so it might not be relevant in the next versions of UE4 and / or the Wwise plugin.
Have fun !