I don't think that Amperex white labels would be overkill. That's the great thing about tube equipment, you can make it sound better by changing the tubes. I have my BAT VK-P5 phono preamp loaded with Amperex 7308 PQ's (white label) and love it. It was a tremendous improvement over the stock Sovteks. I would say that the difference between the stock tubes and the Amperex tubes was bigger than a $1200 mod that was performed by GNSC, thereby being more cost effective.
Soup that baby up and kick back and enjoy!
FWIW, I'm running BAT seperates, and though they are definitely better than the integrated, you have to pay a lot more $$$. Whether it's worth it or not is a personal decision. My seperates run almost 4X the price of the integrated, they are definitely not 4X's as good. The law of diminishing returns are alive and well, just like with other products.