For the next test I removed CR1,CR2 and CR4 (TVS diodes for 5V, shorting to ground in case of excess voltage).
This way I could increase power to the H-bridge powering the peltier elements, and keep still a 5V supply to the microcontroller and ADC.
Attached is the pin out of the original power supply.
pin 1 gives power to the peltier elements, pin 2 gives power to the micro and ADC, 3 to 5 should be tied together to ground.
I tried several runs, first with 6.5V, than with 7V and finally with 8V for the peltier elements. Each time heating up went a little faster, and stability (aka "check ready") was also sooner reached.
Remark: the removal of CR2 is normally not necessary, this one protects the 5V for the micro and ADC, I have later reinstalled it but forgot to take a picture of it.