Although this requires some manual editing, you can try the following.
1. Make sure Origin and NFS Payback are closed.
2. Go to C:\Users\USERNAME\Documents\Need For Speed(TM) Payback\settings
3. Find the file PROFILEOPTIONS_profile and open it with a text editor, Notepad should work
4. Find "GstRender.FullscreenEnabled" and set it to 1. Here is how it is set up on my end: GstRender.FullscreenEnabled 1
5. Save the file. Do not "save as".
Now the game should force open in full screen. Not sure why ALT + ENTER does not work for you, it works here just fine.
In case you are interested, here are my full settings (all changed in-game).
GstAudio.DynamicsEnum 0
GstAudio.MusicModeEnum 1
GstAudio.MusicVolume 0.000000
GstAudio.PoliceScannerOutputEnum 0
GstAudio.PoliceScannerVolume 1.000000
GstAudio.SoundEffectVolume 1.000000
GstAudio.SpeechVolume 1.000000
GstAudio.StereoMixdown 0
GstAudio.Voip 0
GstAudio.VoipCategory 1
GstRender.AmbientOcclusion 1
GstRender.AnisotropicFilter 1
GstRender.AntiAliasingPost 2
GstRender.Brightness 0.500000
GstRender.DisplayResolution_ID
GstRender.EffectsQuality 2
GstRender.FullscreenEnabled 1
GstRender.FullscreenMode 0
GstRender.FullscreenRefreshRate 120
GstRender.LightingQuality 2
GstRender.MeshQuality 2
GstRender.MotionBlurEnabled 0
GstRender.OverallGraphicsQuality 5
GstRender.PostProcessQuality 2
GstRender.ResolutionHeight 1080
GstRender.ResolutionWidth 1920
GstRender.ScreenSafeAreaHasBeenSet 1
GstRender.ScreenSafeAreaHeight 1.000000
GstRender.ScreenSafeAreaWidth 1.000000
GstRender.TerrainQuality 2
GstRender.TextureFiltering 2
GstRender.TextureQuality 2
GstRender.UndergrowthQuality 1
GstRender.VSyncEnabled 0
