I'm intrigued what date you are looking at, as £350 for an LHR sounds off the scale expensive. As to which option makes sense, it really does come down to personal preference. I'd rather press on to LHR and then have a lie in the following morning, but I don't own any children and have no idea how yours respond to late nights.

As for safety - nothing to worry about at all in London. This isn't St Louis!