On the rare occasions I've had to deal with generator issues (not FZ1) it's the coil assembly that's been the problem. That said there was/is an issue with early models of FZ1 08 - 09 in that the magnets became dislodged from the stator drum and stopped charging and/or destroyed themselves, in extreme cases wrecked the motor. Yamaha changed the design of the drum and effectively fitted a sleeve to prevent the magnets from dropping, new part No 2SH-81450-00, old part 5VY-81450-00. Check the part No on yours, if it's the latter, it's very likely this will be your problem and a change of the rotor assembly, could well sort it but wait for it £675
A contact suggested these
https://www.motoelectrical.co.uk/types/stator-coils-/flywheel-rotor-yamaha-yzf-r1-fz1-1000-fzs1-fz8-800-2004-2015/?gclid=Cj0KCQjw3Nv3BRC8ARIsAPh8hgKQrLj_Vuvh7KUwLSg9UWGR86qCXT-WlLi_ibWkhiMVZ6dqGMa_xjAaAiv6EALw_wcB £214
Or I found this
https://www.ebay.co.uk/i/264477816095?chn=ps&norover=1&mkevt=1&mkrid=710-134428-41853-0&mkcid=2&itemid=264477816095&targetid=908661247816&device=c&mktype=pla&googleloc=1006675&poi=&campaignid=10199419183&mkgroupid=101401896465&rlsatarget=pla-908661247816&abcId=1145983&merchantid=6995734&gclid=Cj0KCQjw3Nv3BRC8ARIsAPh8hgIFzxnXZRTgU8dux2SbIC4zs8DFYN09BjfGojbrUjUXUfloSRGiakQaAjBfEALw_wcB Which is genuine and the later part £370
If it does turn out the rotor assembly is the later type, the coil assembly is then more likely the issue, I'd go for either new £355ish, second hand OE around £150 ish.
Or you could go pattern part as with any pattern part I would advise caution, that's not to say they're wont work, they will, it's just for how long and often they don't quite always fit as they should! You've also got wild variation in cost, Wemto part
https://www.wemoto.com/bikes/yamaha/fz1_n/08-09/picture/generator_-_stator is £219ish for what looks like the MPW offering
https://www.motorcyclepartswarehouse.co.uk/mpw-pattern-replacement-stator-generator-assembly-yamaha-fz1-fazer-2006-2012.html which is £46.99 not sure if that's inc VAT but that's a huge difference!
Both are probably Chinese copies, it's your choice.
If you've not removes a generator assembly before, undo the centre cap but don't undo the 3 small bolts around the cap cover. Undo all the bolts around the cover these are different sizes, now use a a lager Phillips screwdriver or Allen key to push on the stator drive shaft. This will allow you the remove the coil assembly while keeping the rotor drum in place, it will be very magnetic and require a really good straight pull. If you allow the rotor drum to be extracted with the cover you run the risk of dropping the shaft spacer and washer into the crankcase. Even when you do it this way there's still a risk, but much smaller, once you've got the cover off, very smoothly pull the rotor drum out, the drive gear should remain in place. Now look through the hole in the gear centre, you should see the displaced spacer washer, using the screwdriver/key insert into the gear hole until it stops, now you can pull out the gear, be very careful as behind the washer is the spacer. you should be able to get the gear/washer and spacer onto the screwdriver/key.