The FZS1000 has the SH650C-11 from Shindengen. This is a thyristor shunt type.
The FZ1 has a FH012AA from Shindengen, a Mosfet shunt type.
Mosefet types are the more modern ones. Most of the RR are almost indentical and interchangeable, only current ratings have to match (or higher). Staying with Yamaha you might have luck with the connector matching.
I have installed a "compu-fire" 55402 on my bike, this a series type regulator. Those help to cool the alternators as the overage heat is dissipated in the RR (they have large cooling fins), not the alternator.
Positive effect of these series regulators on charging voltage is minor, at best.
At idle the FZS1000 alternator is not able to provide suffcient power for all consumers. Regardless whether shunt or series regulator. It needs at least 2000 revs to contribute signifcantly and charge the battery.
If pricing is an issue you can take a used SH or FH regulator, hoping it still works properly.
Next (better) choice would be to look for a FH012AA or FH020AA replacement kit (e.g.
http://roadstercycle.com/). Mostely they come with a cabling/connector kit.
Most expensive are those series type RRs. Company names are compu-fire, cycleelectricinc, roadstercycle or silent-hektik.
Before changing the RR I would inspect the 2 connectors of the 3 wires coming from the alternator. On any doubt I would cut them out and solder the wires directly.