[Doc]

JJ or AK, and I'd be using my observations at the table to decide. Many players won't raise pre-flop with JJ and similarly others will flat call with AK, especially with position. A good player IMO in this situation would raise with either, and probably by a similar amount. With position, 2 callers isn't a disaster with JJ, but you'd prefer none or 1. Similar with AK.

Again, I think a good player should bet the flop. There aren't any flush draws, but an ace, queen, ten or 9 puts some likely straights out there. Also, against two players he has to reckon he is likely up against a pair. Being scared on the flop (e.g. QQ being wary of a King) doesn't seem too likely as he has position and you've both checked. A continuation bet will take this pot down most times.

The check on the turn screams that he has nothing or has something strong - you've both shown no strength so any mediocre hand has to try and take the pot now.

You bet the pot on the river and he raises big, but not all-in. If he was scared of a king while holding QQ etc, this play makes no sense. So he isn't scared. The three couldn't help him - only one left in the deck and he certainly wasn't playing K3. He similarly wasn't playing KJ.
It is a bad move with AK as you could credibly have JJ or 33 (as you do). KJ is also a possible holding for you.

He has a bigger than average stack, so I'll assume he is better than average. JJ it is.