I have come to the conclusion that the DS1307 will not work.
However, the PCF8563 will. This is a very similar chip and rather inexpensive (10 for $8.12 on ebay). It only needs a 32.768 kHz crystal to work. The crystals are on ebay 10 for $2.50 or 50 for $7. So for about a dollar (US) each you have an RTC with interrupt output.
There are two types of interrupts, Alarm and Timer. The timer is a count down timer of an 8-bit register. It can count down at 4 different clock rates but the most useful would be the one second and one minute rates. So in one second resolution you can sleep for 255 seconds (4 minutes 15 seconds). Or with one minute resolution you can sleep for 255 minutes (4 hours 15 minutes).
This a both cheap and versatile.