The resistor in series with the LED should be shown on the schematic, not on a photo.
You say you want the alarm to light and sound when a laser beam is broken.
For how long, what turns off the alarm? Do you need a timer to turn off the alarm after a few minutes if a big truck blocks the beam for weeks? The original circuit's timer does not time-out if the beam is continuously blocked.
Why are you talking about a Fet? A Fet, a transistor, a 555 or an LDR could activate a piezo beeper. Any one of them can be used. If the light beam is dim or if the LDR has poor sensitivity then it must be used to drive a Fet, a transistor or a 555.