potentiometer like rotation angle sensor
If they're available they'll be difficult to find. I suggest that you do some reverse engineering on a working unit. Record the voltages on each tab at 45° increments of the shaft.

Angle | Tab 1 | Tab 2 | Tab 3 | Tab 4
------+-------+-------+-------+------
   0  |       |       |       |
  45  |       |       |       |
  90  |       |       |       |


You may be able to use the results to work out how to simulate the pot with a microcontroller's analog outputs driven by rotary encoder.

Alternatively you may be able to use something like the AlpsLine RDC80 which manages the 360° rotation with two overlapping 340° tapers.