SR access

Is is possible to address 3 SRs individually? Each SR will have 8 LEDs attached. A sample program uses highByte and lowByte but that only covers 2 SRs. I was thinking they could be addressed by activating a designated pin on each SR HIGH that is attached to the microcontroller's output pins. Think that will work?

What is a SR ?

A shift register ?
You can cascade as many shift registers as you like.
You have to adapt the sketch, but that should be no problem, and you can always ask us.
You can use shift registers with seperate enable or clock signals. But in the end, the sketch will not be easier or harder than with everything cascaded.

Yes, you can control the latch pin of each individually.
Connect Clock, Serial Data In of all in parallel.
Connect Latch pin of each to its own control pin.

Just as easy to daisy chain all 3, and use SPI to blast the data out from an array whenever you make a change in the array.

void loop(){
if (arrayChangedFlag == 1){
arrayChagnedFlag = 0; // clear for next pass
digitalWrite(latchPin, LOW);
SPI.transfer(dataArray[0]);
SPI.transfer(dataArrau[1]);
SPI.transfer(dataArray[2]);
digitalWrite(latchPin, HIGH);
}
// do other stuff in loop, including whatever updates the 3 bytes in dataArray.
// set the arrayChangedFlag to 1 when you do
} // end loop