ok well here is what i know on the matter, (be forgiving we are just learning about this stuff now in one of my classes)
a crystal oscillator which are usually made of quarts are very good in terms of stability, they don't drift very far in regrads to time and temperature most other oscillators do.
Now when do you need to use a crystal?
well an LC oscillator (Colpitts, Hartley) use inductors and capacitors, one can mimic an inductor by the use of OpAmps and capacitors. however some times when doing calculators it appears that you need an inductor of a few 100 Henrys, which is rather difficult to make and will defiantly not fit into a watch if you manage to make one.
this is where crystal oscillators come in handy, they can be used instead of a insanity huge inductance.
Now will this work for your application?
I would say not really, you would one need to match closely to what they already have in place, it may work if you can find one, but i view my self as a very practical person who likes to solve the issue the simplest way possible... new clock?
However it would be a fun learning project none the less, but if your goal is to get a more accurate clock, just get a new clock...