keef476:
Is there any reason I should not share the same power supply?
Yes. Inefficiency!
If you are going to power this from batteries, then using the linear regulator in the Arduino (which you can do at 12V if you are powering nothing else from the 5V regulator output) will be wasting 60% of the battery power.
It makes more sense to power the Arduino (and use a Pro Mini instead which has no USB to TTL converter to waste more power) from 4.5V - three AA batteries.
I do have to query why you would be using a "grain of wheat" bulb nowadays when these have been functionally replaced - such as in model railways - by white LEDs in the SMD format which can be powered by the same 4.5V without the switching transistor?