Yes, but Bairstow's form suggests it would be a mistake. He's a very good middle order player and the vacancy would be at #3, which is where Root definitely doesn't want to bat. Foakes is a very good option for a #7/#8 but you wouldn't really want him any higher than that. Maybe Bairstow not having the gloves and instead being Angry Ginger Johnny with a real point to prove would overcome the technical deficiencies enough for him to bat 3, but I just keep seeing him getting bowled by the nip-backers instead.
Lawrence is a very similar kind of player to Pope and someone I thought would be nailed on as England's next excellent Test bat years ago, long before Pope's emergence, because his technique was excellent and he really understood his own game. However it looked like he then had to concentrate a lot on T20 and I think he lost some of what I thought were his real strengths. I don't know if he's successfully merged the two disciplines now, but obviously being a good ODI/T20 batsman is a bonus when it comes to selection for this team.
Definitely someone I really _want_ to succeed, y'know?