I'm sure I will be castigated for this approach and it comes with no guarantees.
I fly to the US on business more than five times a year. If the visitor queues are too long in the arrivals hall, I just join a domestic line (which in MCO,JFK, BOS, PHI and LAX all have fingerprint machines) and "tactically" hide my passport and waiver form. When I arrive at the desk, I hand everything over and smile. It's worked perfectly for the last two years. Only once has a comment about being in the wrong lane been passed which I responded to with a very British "Oh am I?" "I'm so terribly, dreadfully sorry!" Faced with this apparent guilt and embarassment (neither of which is a recognised American trait), the officer quickly told me it really wasn't a problem and processed me on anyway.
There are variations of course: queue with the Domestic's but when released to the mini-lines for the desks, join an international one.
I know. It's not fair and it's not British.