Yeah, that has actually been an issue for us during development. Something like this requires quite a few more assets than a normal webpage, and clearly Chrome's Dev Tools don't handle that too well.
We don't really have a good solution so far. We mainly try to avoid the network tab, but if we must use it, we pause the game to free up some extra resources and generally it won't crash.
Mind if I ask how you managed to solve this? Seems like a painful problem to have!