Second hand in the tournament. You probably don't have any reads. Most weak or lower stakes players are not 3-betting aggressively enough. Especially because of that I agree with you that you don't want to get in 160BB pre-flop without antes. But obviously you have to play queens, so I think calling is by far the best option.
This is in my opinion were you make a big mistake. By check-raising the flop with queens here, you are going to be all-in a lot of the times on the turn or river. Or you will get in super awkward spots out of position. You are super deep and I don't think queens are strong enough. Yes, you might get value from hands like AJ, but I doubt those hand will play for stacks. The button can have AA, KK, JJ, TT, and JTs for value. You will get into trouble a lot of the times. You even block draws like KQs, while not blocking any value hands.
What I would do being so deep is check-raise a narrower range. For example raise JJ, TT, and JTs. Mix this with some bluffs like AQs, 98s with a backdoor flushdraw. Obviously also sometimes call with sets and JT to keep the other player in and have a more balanced range. Playing like this it is very clear what to do on later streets.
If we take the check-raise range we constructed on the flop, we can comfortably continue betting and stacking off with our value hands. The turn does not give us flush draw equity for our bluffs. Maybe continue betting with 98s, and give up with AQs. 98s has the least showdown value. AQs might win against a draw who decides not to bluff.
Now to your actual hand. Despite getting a very good turn, it is getting really awkward. You are playing a massive pot oop and you are unsure where you are at. Frankly, I really don't know what to do here, which is exactly the reason you should try to avoid this spot entirely. Bet-call? Bet-fold? Check-call? Check-fold? I guess I would check call and check fold the river? I am not liking any option. You choose for bet-call, but I don't think the average person is bluffing in this spot all that often. He could have hands like AQs, KQs, 98s for sure. But you showed so much strength it takes a lot of heart to bluff here. He probably wouldn't do this with AJ. I think he is almost always going to have you beat here.
So in short with queens here I would probably check call all three streets, except on bad runouts. Value bet the river if button checks turn.