I wonder if by focusing on the starter circuit because you're suffering fueling issues on start-up that you're missing the more likely cause(s). The starter jets are large bore by comparison with the pilot jets and the whole circuit is hardly fine-tuned ... it just serves to richen the mixture for starting and warm-up really. Witness how reluctant the motor is to idle with 'choke' within 30 - 60 secs of firing up.
Before anything else, I assume that the plugs are good and that you've checked the condition of the plug leads. Always worth removing the caps, trimming 1cm off the lead and refitting the caps into fresh cable cores.
The pilot jets are always suspect in fueling issues. Soak in solvent/carb cleaner and be certain that the tiny longitudenal hole is 100% clear. If only partially clear, this will cause hesitant running until the motor is up to temp.
Replace the O-rings on the float needle valve seats like ogri48 suggested. They will be in dodgy condition on almost all Fazer 1000s these days. If they aren't leaking already, they will be soon. Any leakage past them will alter the fuel level in the float bowl and mess with the fueling. It is possible that a high fuel level is causing a rich misfire on cold start up but once the motor is running harder, the demand for fuel is sufficient to reduce the level towards its normal level. This would account for the rough running disappearing when the motor is up to temp.
Do the preceding and then run a few tanks of fuel through with carb cleaner additive to reduce any varnishing that may have occurred in the internal passages. Ultrasonic cleaning should have the same effect but will be far more costly.
FWIW, I've seen Fazer carbs internals that looked diabolical (like green algae, almost!) but they still ran reasonably well. I doubt yours are so badly gummed up as to be beyond sorting if you follow the advice given so far in this thread.