Competitive play requires time to think about a move negotiate trades and then act on a plan and a 2 minute turn timer just does not allow for that. Instead I propose a turn timer similar to how it is done in chess where each player has their own timer. I would say each player should get 30 minutes the guarantee the game last no longer than 2 hours which I think is a respectable length for a competitive game. In tournament I have seen games last longer. But it can really be any amount of time. When a players turn begins their timer starts counting down and it ends when they end their turn. Timer also runs during discarding. If a player exceeds their timer they lose a point and gain a tenth of the timer back. So if the timer was 30 minutes and you exceed your timer you lose a point but gain 3 minutes in your timer.
This allows a layer of time management to the game but also doesnt punish a player for having a lot of cards in their hand and having a big turn.