Will it shift into 2nd with the engine off? Hopefully it's just a clutch. Roll it forward while trying to shift into 2nd with the engine off, if it goes in while off but not on it could just be a clutch. The bike has to be moving a bit to go into 2nd from 1st with the engine off. 2nd gear issues are very common with a lot of 600cc sport bikes, the engine in the FZ6R i believe is just a differently tuned older R6 engine. They are known to have 2nd gear issues, typically caused by missing second a lot and/or clutching up in second to do wheelies or hard pulls. I'm certainly not a transmission expert, but I can tell you whats involved in replacing a transmission on a motorcycle like yours, I ride a 05 ninja 636 which is also plagued with 2nd gear issues I have looked into it. You have to split the case of the engine to even get at the transmission, it's not like a car where the transmission and engine are in separate housings, on a bike like yours a good bit of the engine has to be taken apart to even get at the transmission, its very involved and likely beyond the skills of your average DIY. To have say shift forks, gears, and whatever else you need replaced in a shop will be very expensive in just labor costs not including parts, if I had to guess over $2000 depending on the parts that need replacing. Unless you plan to do the job yourself it would almost certainly be cheaper to just replace the whole engine with a good used one.
