How to supply 2 different voltages to a circuit every 2 minutes
Use the solenoid to select between the two input voltages (11V and 14V).

Edit3: I would avoid putting 14V on the output of U1, the 11V stepdown, as shown in the questions schematic. There is no information about U1, and it may survive, but I would avoid doing that. I would expect the stepdown to be happier with an open-output. If that is still a concern, add a resistor between the output of "U1 STEPDOWN" and ground to keep it within spec. At a guess it might need a few mA, so a 1K resistor may be enough.







simulate this circuit &ndash; Schematic created using CircuitLab

The transistor switch could be a N-MOSFET, or a BJT with current limiting resistor on the base. 

Edit:
The N-MOSFET (or NPN BJT) could be driven by any square wave, it doesn't need to be an MCU.

In favour of a low-speed oscillator is simplicity and cost.
In favour of an MCU is flexibility. For example a sub-£3 Arduino could monitor some aspect of the circuit under test, and capture data, or be reprogrammed to have more complex timing.

Edit2: tidied up schematic, hopefully addressing concerns.