I see no problem doing that, you could either do them both by polling, both by interrupts, or one of each.
As I recall you can set the PCF8583 to output a pulse every second on one of the pins, you could use that to signal when the next update is needed.
Or you could simply read the clock during the half of the pulse you're not reading - if you're measuring the LOW part of the pulse, use the HIGH part to read the clock.
You might like to consult the (very old) MicroChip datasheet for a frequency counter, it does much of what you want - while multiplexing LED displays at the same time. There's a later modified version (not from MicroChip) which uses an LCD display instead.