IMO the best are the dense crunchy knits from that one German factory. You can get them at Hermes for $200+ or, when they have them, from Conrad Wu or Kamakura for about $70. If you don't see them IRL, you have to look for the made in Germany label.
My tux is a black barathea from Scabal. It was scratchy in the beginning and softened with wear. I don't know the cloth you bought.A popular alternative (on SF) is Smiths wool/mohair blends. I have midnight blue tails in Smiths 8810 and it feels lighter than my tux.Here is a picture of foos tux in 8810.
The basted fitting is key. Every cloth behaves differently. I can't believe you guys skip that especially when you know your shoulders are asymmetric. You can get them to try to fix it after the fact, but I suspect they can do much more adjustment at the basted fitting stage.
Aren't you from Texas or somewhere else from the South? If so, you should consider linen or an open weave cloth. If you were looking for a suit, I'd recommend mohair - scratchy but performs unbelievably well in high humidity.