I don't see why you need all those extra parts. 13 pins is plenty for what
you said. The keypad and LCD and 3 LEDS and even the piezo buzzer can
share pins. I bet you can do this with 12 pins total, including your 2
wire comms. You don't need a shift register. Your LCD accesses are fast
enough the LEDs will barely tickle. Or put a small cap across each one to
get rid of that. The piezo should probably take the 13th pin though. Even
if you use the LCD register select pin, the piezo make tiny noises that
will be audible.

Cheers,
Bob