I think that the UK's natural home would be more with the US than with the EU. If the UK is going to incorporate with another entity, it would do better to become a US state where it could weild considerable, very considerable influence. Already, early US history is dominated by British history. When learning about US government in school, the first several chapters are all about Britain's great strides towards democracy and the eventual transcendence of that british endeavour into our own form of government. Today, more people of British origins reside in the United States than anywhere else in the world. For Britain to join with the United States would be to "re-unite" those people into the Union.
So, I definitely think Britain would do well as a free & independent state in the US. It could easily escape federal regulations by stressing its rights as a free and independent state. British admission into the Union would extend the size of the House and Senate and provide a positive influence on Presidential races.
Also, British culture would be amplified through the influx of support from the American media machine and the federal apparatus would propel Britain into a global superstar outmatching even the cultural influence of states like California, New York, or Texas.
Restore the Union! Statehood for GB!
GB leaders of the 20th century such as Blair, Churchill, Thatcher, etc are all extremely revered in the US. Our alliance in WW1, WW2, Cold War has done created an atmosphere of Union between the two. GB need not be America's "lapdog" but rather, it can be its leading force by re-uniting with the Union.