@berlinonout suggests making an extra notch in the original 2-speed shifter to make it shift over the 3 positions. In my opinion this is not practical – you will likely just damage the shifter. If your Brompton is pre-2017, you then need to reverse the shift cable’s direction so that its stop ends up on the side of the shifter, when going with a standard shifter such as SunRace. There are barrel nuts offered in the market for the purpose, but the most economic solution is just to get derailleur anchorage for the new Brompton shifter that relies on the standard shift cable orientation. As this new anchorage is a hell to set up, though, as already mentioned in another thread, you may just as well fall onto the easier barrel nut.
Great post! Small error: I did not suggest making an extra notch - I mentioned it as one possiblity, especially if you want to safe on cost. I am using a Sunrace thumbshifter on my build (and pictured that as well) as I like this better and the newer Brompton dogleg instead of your way with the barrel nut as it is cheap and works very well. Did not find it difficult to set up.