ICSP + SST25VF016B + ATmega2560 Problem
The bomb-proof method of doing this is as below 

When programming via SPI (I'm assuming you're using AVR-ISP or the latest version of it to do the initial production programming), the unit is held in reset. So, this method isolates all SPI from the processor other than the programming unit.

Be careful with the reset chip, I've been caught out by a push-pull output that takes control of the reset line and have got over that with an open drain device or a series resistor between output of reset chip and rest of the circuit. 

You can use a 4053, doesn't have to be a 74HC.