It's pretty simple to do a full clock calendar as well, but that didn't seem required from what he's said.
In fact there's even a MicroChip application note that uses a PIC as an I2C slave clock/calendar - sort of a home made DS1307
As far the DS1307/3231 goes, you could obviously use that in conjunction with a processor, but for simple once per second toggling there doesn't seem much point?. What I usually do with the DS's is set them up to output one second pulses, and use that as an interrupt input to the PIC (which then would read the DS in the ISR and update the display) - you could use that interrupt to toggle the I/O pin as I suggested above. However, it seems a waste of a DS chip?.