Sadly, this will work on some people - they've been heavily conditioned to feel ashamed and to respond to that shame with compliance.

Those people will virtue signal and/or parrot the narrative and push the overton window that little bit further toward 'It's totally their business to censor your games due to concerns about the fictional content you engage with in your home! You're not a war crime apologist, are you?'

Unless there's a counter-narrative from someone high on the progressive stack who can claim 'This campaign is harmful to people like me', in which case they'll bitch and whine enough that the campaign will be rescinded and apologized for.

Or it could just fizzle out without gaining traction. The Red Cross might not be sociopolitically relevant enough to introduce bills at the Progressive Extrajudicial Legislature.