Personally I can't see spending the up to $200 or more that the typical "Ultra Performance" tire costs in that situation. However, if you looked at my second post after I had gone to the Tire Rack site and the tires ckj688 has mentioned are well under $100 each right in the same price range something like a BFGoodrich Radial TA and suggested that he should go for it.
Then after seeing the upgrades he has down to the suspension the tires are right in line with what would be appropriate.
As far as my being able to get the full potential out of my 360, I do almost every time I drive it. I have done a lot of suspension upgrades, to my car not as extensive as the coil over Alterkation kit cjk688 has done but I have made a very significant improvement in the handling.