MN1307 just sits in a loop continuously reading the registers of the DS1307 chip (about four times a second) and displaying the results.

If time is going by very slowly, then it is nothing to do with SDA/SCL, LCD or VBat. Either you have a defective DS1307, or a bad 32kHz Xtal (most likely), or you have additional Capacitance on that xtal that shouldn't be there... however to slow it down that much, I wouldn't think any Capacitance would pull it that slow.

Rip apart a cheap Digital watch, steal the xtal, it should work with the DS1307. Even an xtal cut with the wrong Capacitance will work, though it keeps rubbish time (ie could be out by as much as 30 minutes in a year).