Why did I fry my lab power supply trying to charge a lifepo battery?
Mea culpa, well, sort of. The flyback diode was dead (V32 on the schematics - no fuse indeed), and it was short-cutting the output, hence the smoke also above in the circuit. I replaced them and it worked again.

Now for "why": I have two of the same 20AH LiFePo battery. And one packaging is plain wrong! Yes, the plastic wrapping with the label is the other way round: inverted positive and negative terminals.
I had probably double-checked with one, and assumed that the other was alike. I am going to yell loud at the supplier (a well-known company though). It could have cause fire, with such currents. Amazing.

Conclusions:


One can never check enough the polarity (batteries were alike, I tested the right one, and probably plugged the other)
Do it one at a time, it would not have occurred
Do not even trust the labels and packaging! :(
KISS: when the clamp diodes are dead, a reverse polarity is the obvious thing to check first (so @Bradman was right indeed)
My power supply works well again, warranty void (bah)


Bonus: I replaced the SMT diodes (1N5402) by even more powerful diodes (31DF, 300V,60A peak) + fixed them with silicone so they would not even unsolder themselves on a terrific case (like here I guess).