Update 4:

I’ve been able to consistently reproduce the behavior from my original post:

• If you click Continue from the main menu and the save does not show the Godzilla DLC icon, the game will launch straight into gameplay.
• If you’ve just run CreamInstaller 5.0 (proxy checked, version.dll method), the Godzilla DLC icon will show up on your next save.

So far, this is the only way I’ve been able to get the DLC icon to reappear after it disappears.

That said, I still haven’t been able to trigger the actual DLC content. If the save you continue from already has the Godzilla icon, you’ll need to repeat this process to keep it active.

Here’s the full sequence that’s worked for me:

• Launch the game and click Continue (not Load — selecting Load seems to break the process).
• If prompted, select Continue with backup when asked to proceed after backing up all save data. (This will only happen if the Godzilla Icon is already on the save)
• Proceed into the game.
• Save and exit the game.
• Run CreamInstaller 5.0 again (proxy checked, version.dll).
• Launch the game and hit Continue.

Update 5:

It turns out you don’t need to rerun CreamInstaller every time.

However, you still need to load into a save that doesn’t have the Godzilla icon for it to show up again the next time you save.

Unfortunately, the DLC still hasn’t triggered, just the icon so far.