Interestingly, looking at "Manage Your Kindle" at the UK store reveals a big banner saying "Your digital account is currently registered at Amazon.com. Go to the Manage Your Kindle page at Amazon.com." Doing the same at the Japan store...just shows me my Japanese "Manage Your Kindle" page like normal. Are the US and UK stores tied together more closely than US and Japan are? Is the UK store stricter about these things?
That would mean any advice I might give might not be very useful, being based on my experiences using the US store while living in Japan. But given that...
Do you really need the proxy? I buy books from Amazon.com from Japan sometimes, and they don't seem to care all that much. Once you register some address in the country you can set that in the Country Settings page of Manage Your Kindle. My old address in the US was a family member that I used to regularly ship boxes to, but they've moved since then so as far as Amazon knows the current address is "untested." So probably any address will do. Perhaps avoid Buckingham Palace...
Gift cards from the US and Japan are totally independent, so I wouldn't expect to be able to use a US card in the UK.
Other people will no doubt have more experience with the UK store than me.