Did exactly this, a few weeks ago. Was heavy Chrome user, now fully on Firefox (and Safari on mobile). I am so satisfied with this decision, and the transition was much easier than expected.
While I run into this too from time to time, it's not something everyone uses. Also it takes me 5 seconds to workaround by debug logging incoming ws messages. Most web devs I talk to can do everything with FF devtools more than fine.
My solution is to use chrome when faced with that particular limitation. There's a bug open on the mozilla tracker and it's been there for a while but no one seems to care.
I use both, Chrome for client side dev, Firefox for everything else. Chrome is just the dev tool. Works like a charm because switching between tabs inside a browser is just as quick as alt-tabbing between browsers.