After almost 16 years of owning a number of Michelin Primacy MXV4 and then MXM4 I've always gone with a 97V load/speed rating for some handling capabilities, but the tires do ride somewhat firm. No more Primacys my size these days.....However would going for a new set of Grand Touring tires with a 97H rating (with the somewhat softer sidewalls) affect much handling and possibly add a somewhat softer ride for my barge?

 

Honestly, like you said, it's a barge. I doubt you'll notice a huge difference through all the damping that these cars have. I think you'll see slightly reduced NVH levels, but tire pressure will affect that as well. Softer sidewalls may make it feel a bit more squishy or wiggly if you're pushing the car hard and fast, but... if you're up for a hard and fast drive, it's probably better to take your other car.

Interesting side note, only US models can wear H rated tires since H is rated for 130mph and US spec cars have a 131mph speed limiter, while the Euro and JDM spec cars have to wear V, ZR, W or Y rated tires because their limiter is set to 155mph.