Reddit user Ian_Ian1 found this fix:

Steps: Using ProtonUp-QT, download GE-Proton7-50

Click Properties, under compatibility, force game to use GE-Proton7-50

This will allow FMV to play but NO AUDIO

3. To fix audio, go under shortcut in the same window. Paste this launch option=

GST_PLUGIN_FEATURE_RANK=protonaudioconverterbin:NONE %command%

This will enable English audio to play in FMV

If your audio on the opening screen is still stuck in Japanese, you need to delete the SYSTEM config save file to reset it back to default audio language which is English.

Location: /home/deck/.steam/steam/steamapps/compatdata/1580780/pfx/drive_c/users/steamuser/Documents/KoeiTecmo/NINJAGAIDENSIGMA/user_id/SYSTEM

BONUS

I tried the same thing on Ninja Gaiden Sigma 2 + NINJA GAIDEN 3: Razor's Edge.

The opening FMV+audio plays but I cannot guarantee English audio works as I have not progressed far enough to confirm it myself. Leave a comment if it works for you :)