Carb Fazer 600's have always been tricky starters, it's a balancing act of choke and throttle, many of the last crab bikes where/are my XJR for one.
The choke isn't a choke in the sense of the word as it doesn't add any fuel just cut the air which has the effect of enriching the mixture. It's an enricher circuit.
Fazers run rich slightly rich at the low end as standard which is why they respond well to a high flow filter and end can without jetting. With a standard can/filter I've had some success with setting the air mixture screws at 1 3/4 and 1 1/2 turns out but you need to test to ensure you'r not leaning off too much. Most often than not you don't need any throttle or choke to start the bike unless it's cold, it's really a matter of getting used to what works for your bike. Mine is about 5mm of choke and no throttle, let it run until the rev start to rise and back off the choke slightly.
I suspect the issue of starting lies in the ignition, something like a 4* advance would help, it certainly does with the XJR, but I don't know of a advancer plate made for a Fazer. That said it's possible someone makes one, if they do you could try it.
Without a plate it would be a real fuff for very little gain, IMHO just learn what works to start yours.