Exact (accurate) and cheapest Second Signal Generator circuit
It looks to me like a CMOS 7200 wristwatch chip would do what you need. According to the CMOS Cookbook, if you connect a 1.55 volt input through a 10k resistor into the TEST pin on this chip it will put out a 1 Hz square wave across the resistor. That's intended for calibrating the time, but in your application, you could skip all the other hookup to the watch and simply wire the chip to 2 1.5 volt button batteries, in series for the ~3.1V main power, and the center tap of the 2 batteries to the TEST pin. 

You could then put a transistor across the 10K resistor to boost the voltage drop across the resistor to +5V, and feed that into your watch control circuit.

It looks like a TPL5000DGST programmable timer would be even better. If I read the data sheet correctly, that is an extremely low-power timer chip that is designed to output pulses at a settable interval where 1 HZ is one of the options. It looks like that chip includes a built-in oscillator. You could feed it regulated 5V input and you should get clean logic-level outputs out of it.

Disclaimer: I'm a software developer, not an EE. I'm getting this info by reading books and data sheets.