Exactly! When I have to speak with actual English people, I do try my best to imitate a Americanised, TV show accent. When I speak to non-native speaker, I don't try and let my french go through. It's easier for everyone.

At some point I started to embrace my rolling Rs, "ze" all the way and rhyming passage and massage. But luckily I live at the bottom of the sea, where everyone is an English speaker, but nobody is a native.

I'm the other way around. I sound like a native US English speaker, but when I'm speaking English around French people who aren't as fluent, I "Frenchify" my accent so it's easier for them.

My spouse finds it amusing, which is probably the biggest benefit, TBH.