To get useful data points it's helpful to look at what the fuse will likely do from a datasheet:
At the standard temperature (usually 25°C) and 1A, a 600mA-rated fuse will typically blow in about 15 seconds. The time will be less if the ambient temperature is higher, and greater if the temperature is cooler than 25°C, but even a +/-10°C ambient change will only change the effective current rating by about -/+5% (see below).
You can measure the current through the light bulbs without the fuse and see if you can get current values that make sense from the perspective of the fuse characteristics. Putting an overvoltage on an incandescent lamp reduces its life by something like the 13th power of the voltage.. so you may burn out the lamps (keep in mind that the fuse element is thicker and will respond more quickly than the fuse) but you may be able to get useful results. Even if the life of a 1000 hour bulb is reduced 10,000:1 it will still operate for a number of minutes.
Currents in the 1A range are more than what a PP3 9V alkaline battery
may be able to supply, and it may significantly reduce the output
voltage of even a fresh high-quality AA alkaline cell.
The lamp filaments will increase in resistance when they get hotter,
so with over-voltage they won't give you as much current as you might
If you want to do a useful experiment, you should monitor the current (for sure) and also the voltage.