Mak - I'd say go the reciprocal licence route if you have time to get the CAA authentication and get checked out solo while you are there rather than just 'flying with an instructor'. If you buy the FAR/AIM (and "VFR radio proceedures in the USA") books before you go and read them on the flight there you will a) be less bored on the flight and b) know enough to get a BFR when you get there.
You don't need a visa for a checkout or BFR and they are often combined in a single flight depending on your experience. Flying in the US is actually much easier than you might think (and much more rewarding) and once you have the recip, it's valid along with your UK licence.
FFF - sorry to bang on about this, but I note you use the phrase 'formal training'. Do you think there is any accepted definition of what constitutes 'training for licence / rating'?
For instance, if you ride right front seat and the instructor drives the plane, you are just a passenger and I don't see how this could be construed as 'training'.
But what if you sit in the left seat? Does that constitute 'training'? Is it relevant whether or not the instructor endorses your log book, or if you claim the time yourself?
David