The FZ1 has a close ratio transmission, which means that the lower gears such as 1st, are turning slowly at slow speeds, and not able to mesh so easily. Therefore you should downshift at higher speeds than on other bikes. I have found that as you approach a stop sign, start downshifting at about 30 MPH, make each downshift with authority, and then release pressure on the shifter. Proceed to the next gear, etc. The last downshift into 1st gear should be done at no slower than about 15 MPH. If you wait until 5 MPH, then you will experience the problem you described.
At that point you have to ease out on the clutch lever, downshift, ease out again, downshift, etc. until in first.
Get in the habit of downshifting earlier, and I believe the problem will disappear.