Circuit to delay sensor output
A simple suggestion:
The same Arduino code that you are using now can run in a stand-alone microcontroller, instead of running in the microcontroller of the Arduino's board.
Perhaps it won't even be necessary to use a crystal, or other external components for the microcontroller.
The choice of the microcontroller is linked to the complexity and size of your code, and the number of pins of Arduino that you use now.
You can use the microcontroller ATMega328 (the same of Arduino Uno), or even, if your code is small enough, and the circuit use just a few of Arduino's pins, you can recompile your code and run it in a simpler microcontroller, like ATTiny85 (a small IC with just 8 pins).