OK, I appear to have solved the shift to middle cog problem and the main culprit turns out to be pretty embarrassing: clogged up derailleur pulleys and especially the upper one. After cleaning up the tensioner in an ultrasonic cleaner down to bare plastic the problem is completely gone.
On the way I also found that the axle bolt for the pusher was loose and that the coupler between the shift cables from the derailleur and shifter was sticking. That new anchorage basically runs a second shift cable from derailleur, with an orientation reverse to that from the shifter, and couples them. For the moment, to solve this I just went back to the old anchorage.
Finally I tightened cogs on the cassette, going with a spacer from a 9-speed cassette and 12t cog also from 9-speed and just filing them down a bit. I.e. the news is that you can use 9-speed pieces on the Brompton cassette just as long as you use a thin retaining ring.
I still have some unease in the drivetrain (momentary skating) when running at the highest gear ratios, but this is somewhat normal when ironing out the wrinkles. An update to the gear chart from @
pastorbobnlnh/Sheldon Brown is below, I incorrectly claimed that my middle ring is 34t - it is 32t. (It was 34 on the way, hence confusion.) I use distance traveled because I understand what it means. However, Sheldon's calculator gives too few digits for the lowest gears. In terms of practicality there are 15 distinct gears here.
Chart for 27-Speed Brompton