It is not legal under FIS rules, and unlike many FIS rules, their reasons are pretty good.
1) if any of your fiber switches or media converters at either end lose power, you lose any impulses while the fiber is dark.
2) Fiber devices typically don't transmit circuit closures (ie gate-open impulses, photocell occlusions) down the fiber. The typical devices at either end are packet switching & relaying devices. You'd need to either build or source a device pair that would transmit a signal down the fiber indicating the circuit had closed. I've looked at some of these (the ones I've found come from audio) and none of them have any sort of time-reliable or thermocompensated circuitry. So the delay between when the start gate fired and when the impulse arrived at your timer would be random.
The technical hurdles are unfortunate, because these days (at least at the World Cup and World Championship level) an increasing number of hills have SM fiber from top to bottom for HD cameras. The fiber is impervious to electromagnetic interference from things like sound systems, conductance, and inductance, and of course SM reliably transmits 100 Km or so. Would be a good solution if some of the technical issues could be obviated.
