Hello again.
I tried my idea. There is no problem with setting voltage, but I have problem reading it back (see the picture). Any current drawn by the floating circuit goes through the shunt (because floating ground is connected to the output which is happen to be shunt). So, even if I buffer output voltage (in this case it will be virtual ground), the current drawn from the sensing circuit will go through the shunt (the return path).
There were several proposals using differential amplifier. My reasons to avoid howland/differential circuits were:
1) they have low input impedance
2) CMRR is poor and I need to match resistors
3) limited output swing (afaik)
Now I'm thinking that it might be not so bad idea, esp. because common mode shouldn't be too big across the shunt. So, I'll try circuits posted earlier.
There are ready instrumentation amplifiers, but they are quite slow, I'm worried this may affect performance. They are also mostly low-voltage (3-7V range typically, afaik).
PS emitter follower vs common emitter: it's an interesting detail that I missed originally. This explains why moving reference voltage from positive output to ground make the circuit unstable, especially when switching CC/CV mode. I spent a lot of time stabilizing it, and mostly failed. So, good to know.
PPS I also tried doing some analog switching with jfets and low-level mosfets, but this didn't get me far.