If you could hear me guest, you'd hear a very hollow groan, lol - do they have huge root balls, well if I tell you that it once took me 5 hours to dig one out (I was fit, young and strong then) just to dispose of it, which means much less care with the rootball, that might give you a clue. Couple of strong, patient men and you might succeed - good luck with that.

With a pickaxe and a chainsaw you can hack out a sideshoot or two to transplant elsewhere, but this is best done in spring. Water well and it will grow away, so you can have the problem in two places instead of one!
As Bamboo says, you can dispose of the original with a small thermonuclear device or a stump cutter hired for a day.
Seriously though, do consider other types of grasses as there are so many available nowadays. Even big bold types like Miscanthus Giganteus are less invasive and overwhelming than Pampas, and are very attractive when flowering in the autumn with tall slender tassels. Google 'perennial grasses' and you will find many attractive ones which are better than pampas. You could even plant several types in the new spot, with something wonderfully attractive like Stipa Tenuissima Pony Tails in the foreground, and taller types behind it.