I was on La Sarthe lobbies for most of the day, using the same car. BoP'd (although iirc it's just 100%/100%), no tuning, and with fuel/tire degredation. The best lap I was able to pull off was a 3:26 on RHs.
Some people were able to pull a 3:21 or so in an Audi R18 '16, so a time within that range is certainly possible for the Toyota
As for the 3:14 run... well, the leaderboards last time only had a 3:16 (in the Nissan GT-R), so maybe with RM or RS as well as an alien it's possible for the Toyota to do it
Edit: After watching Kobayashi's lap, I have a few remarks. First, he seems to be lifting (if not braking) way earlier than he needs to (or at least what GTS leads us to believe). At the Muslanne corners he lifts by as much as a whole meter board away from the actual braking point. Hmm. A wild guess is that he's trying to regenerate as much of the hybrid system as possible? Maybe we can apply this driving style to GTS as well. Another thing is that his car jerks from understeer to the the trajectory that Kobayashi wants it to be once in a while. Judging from this, I'm guessing that the grip levels of his tires are somewhere around RS and RSS, given that those tires can give the car those jerky characteristics