Voltage drop when switching FET
When I try to switch it I get a voltage drop as can be seen in the
  scope picture and causes my microcontroller to reboot (&lt; 5V). I was
  thinking of using a capacitor to filter out the voltage drop.


Good idea! 

Put a diode before the capacitor and then the lamp won't discharge the capacitor when it drags the supply voltage down. Like this:-







simulate this circuit &ndash; Schematic created using CircuitLab

With this arrangement the capacitor only needs to be large enough to power the controller during supply voltage sags. Assuming the regulator needs 7V and the capacitor charges to 11.4V (12V - 0.6V across the diode), you have 4.4V of 'headroom' before the regulator starts to drop out. The formula for capacitor charge/discharge is t = C*V/I, so if the controller draws eg. 20mA then it should last 0.00047F * 4.4V / 0.02A = ~100ms.