I got a Newport 3040 temperature controller and I have trouble setting it up.

I am trying to make it work with a cheap car thermoelectric cooler, which I rewired directly to the controller.
The issue is that it does not stabilize. And I don't have experience with thermoelectric controllers. I need some help understanding whether it works as intended or something is broken.
I found user manual here:
http://www.iproweb.fr/test/0%20-%20MATOS%20GUILLAUME/NEWPORT%20-%203040/DOC%20US.PDF3.4.2.4 Gain
The Gain function controls two parameters of the hybrid PI control loop;
proportional gain and integration time.
When the actual temperature and the set point are different, an error voltage is
generated. This error voltage is directly related to the difference in the actual and set
point temperatures. The error voltage is then amplified by the proportional gain.
This amplified error voltage controls the amount of current driven through the TEC.
The higher the gain, the more current will be driven for any given temperature
difference, with the maximum current being determined by the current limit.
The error voltage also drives an integrator. The integrator’s output also controls the
amount of current being driven through the TEC. The integrator is an amplifier
whose gain is proportional to time. The longer a given error voltage is present, the
more current will be driven through the TEC, with the maximum current being
determined by the current limit. The speed at which the integrator’s output increases
is the integration time, which can be “Slow” or “Fast”.
The allowed Gain values are: 0.2 Slow, 0.6 Slow, 1 Slow, 1 Fast, 2 Slow, 3 Fast, 5
Fast, 6 Slow, 10 Slow, 10 Fast, 20 Slow, 30 Fast, 50 Fast, 60 Slow, 100 Fast or 300
Fast. The number actually defines the proportional loop gain. The slow/fast suffix
indicates the speed at which the integrator’s output increases. The slow setting allows
for larger masses or greater distance between the sensor and the thermo-electric
cooler by slowing the speed of the integrator.
Both the proportional gain and the integration time must be matched to the thermal
characteristics of the TE cooler and sensor. If the settings are incorrect, the
temperature set point will take an excessive amount of time to settle, or it will
oscillate around the set point and never settle.
The Gain setting depends on the type of TE cooler that you are using, but we can
suggest guidelines for selecting the proper gain. Set the gain to 1 fast and increase it
until the actual temperature oscillates around the set temperature. Then reduce the
gain to the next lower value.
The in Principles of operation on page 32-33 (42-43 in PDF) they say:
4.2.7 Proportional Amplifier and Integrator
The proportional amplifier is part of a digitally controlled gain stage consisting of the
analog switches and their associated resistors. The analog switches vary the ratio of
resistance in the feedback circuit to change the gain.
The signal from the difference amplifier is sent to an integrator which reduces the
difference between the set point temperature and the actual temperature to zero,
regardless of the gain setting. An analog switch discharges the integrating capacitor
whenever integration is not required to prevent unnecessary difference signal
integration.
So my guess is that it is actually a digitally controlled fully analog device.
I.e. once the mode is set, the rest is done on the analog side. Digital side at this stage
only does monitoring.
I set up a script to try all the 16 modes for 5 minutes each.
Room temperature is about 25C, target temperature 30C.
Temperature and TE current values are from the controller.
I just checked that it measures current correctly. So the values seem to be real.
Here is what I got:

What surprised me is that it continues to drive the TE at nearly the full current until after target temperature is reached and then tries to go in the opposite direction at full power. Here is the central part of the graph above:

The “10” mode is interesting because it offers both “slow” and “fast” integrator modes. So we can compare them:

Actually I can see that there was one case, "60 Slow" which has decreasing amplitude. But I don't feel that the stabilization time is usable:

So my question is whether this device is working. And if not, what could be wrong with it?
Upd: I had to re-upload all the images to make img tags work.
