Can I use a 12V switching power supply to power a G4 LED bulb?
Can I use a 100-240V => 12VDC (switching?) power supply to power an
  LED bulb?


Maybe. Yes with the correct LED(s).  

NOTE:  An LED is not a "bulb". You cannot look at an LED as a little  incandescent light bulb, they are very different. 




  Is constant current incompatible with switching power supplies?  


No.  A constant current supply can be a switching power supply.  Power supplies can be constant voltage or constant current. For you it is probably best to use a typical constant voltage power supply and a resistor to limit the current.  

It is best to use a voltage just over the LED's forward voltage.  





Example:  A 12V power supply would work well for a 12V Cree XHP50.2 LED
You may be able to find one (or similar) mounted to a "star" PCB.  Try searching for "flashlight LED PCB".



These LEDs get very hot so I would start out trying for for about 11V and 350 mA and see if it gets too hot. Temperature should not burn you.  You should be able to touch the PCB for at least a few seconds.  

For this Cree LED, try a 1 W, 3 Ω resistor.  This should be plenty bright. It would be better to use multiple LEDs at a lower current (higher resistance) to keep the temperate low rather than run this LED at a higher current/temperature.  If you need it brighter or it gets too hot.