First: the kettle will almost certainly work just fine. Some 240V countries are 60Hz, and nowadays manufacturers like to sell their products in lots of places without making multiple versions. So there's a slight chance that electronic functions might behave slightly off in timing, but the heating element won't care at all. From the perspective of the kettle, there's no difference between your split phase system and a domestic 240V system. It's still line wires and a ground. There is no neutral in your scenario, and nowhere to connect it on the kettle anyway.
Second: no you cannot legally install a British receptacle in your US house.
Third: it really doesn't matter that you can't install a UK outlet. Code doesn't have any say over what you plug in to the wall, so you can replace the plug on the kettle with an American standard one and install a matching American outlet. I'd go with the NEMA 6-15, or perhaps a L6-15. The advantage to the NEMA locking outlet is that it makes it really clear that the outlet is dedicated to the kettle - a 6-15 looks quite a lot like a regular 120V outlet. Since it's in a kitchen, you will need a GFCI circuit breaker as well. Once you have the outlet installed and plug changed out, you're good to go. As a bonus you can use a proper four-slot toaster on the same outlet as well, rather than an underpowered American one.