That idea will work, but Q2 will need more than 1 mA base current.
The 2N4403 (the Fairchild one, anyway) is specified with a minimum h
FE (current gain) of 100 at 150 mA collector current, and only 20 at 500 mA collector current.
The typical characteristics graph shows that typically, the gain at I
C = 350 mA is about 70% of the gain at I
C = 150 mA.
So you should be able to see that the 2N4403's current gain could get pretty low as I
C gets up into the hundreds of milliamps. This is a big advantage of MOSFETs and it's why I would use something like
this one instead of a 2N4403, especially if there are only a few in the circuit.
If you really want to use the 2N4403, have a look at the V
CE(sat) specification for 500 mA collector current. They specify a base current of 50 mA! That corresponds to an R2 value of about 82Ω! And the 50 mA current is wasted. But that will work.