A famous bloke once said (probably in Hebrew, written up in Aramaic or Greek and badly translated into middle, modern and somewhat strange English): "Pay unto Caesar, that which is due to Caesar and pay unto God that which is due to God" or words to that effect. Substitute IRS for God and you have a suitably cynical aside!
If your firm over here is good enough and you are reasonably settled then why not go all in? I'm not sure what the rules are on reconnecting with the US if you have formally renounced citizenship. However, I'm sure a birth cert from Yeehaw, TX or wherever will go some way. Even so, UK citizens are generally welcomed in the Land of the Free. An ESTA (great scheme IMO) costs about £20 for two years coverage - more than enough to visit the relos.
I know from second hand experience that our bureaucracy is just as convoluted as any other and it will take roughly one to two years (that was about 10 years ago) to get citizenship and probably involves a test that I would fail!
You might like to look into where your company is actually registered. A Ltd is not you - it is a separate legal entity and the UK (etc) is quite a complicated setup within quite a complicated environment. You might explore Jersey and the Isle of Man for places to headquarter. Don't overlook Eire either (EU with UK to the north, with special arrangements)
Anyway, the snag you have is solely US based and not UK (for a change).