I would think you could. The bulb doesn't know or care what its source of current is. As long as you can supply 2.5 to 3.5 volts at 1.8 amps, you'll get plenty of light from the Stinger.
I built a Li-Ion pack using two 18650 cells in parallel. And two 18650 cells are exactly the same length as three sub-C cells. The advantage here is that the Li-Ion pack has a capacity of 3700mah, a little over twice the Nicad's 1800mah.
But I did have a problem with the Li-Ion protection circuit. Although it supports the Stinger bulb's 1.8A running current, it won't support the start-up current. (An incandescent filament's cold resistance is only 15-20% of it's operating resistance. So all incandescent lamps draw approx. 5 to 7 times more current when they first turn on). This warm up period is very brief, on the order of milliseconds. But the protection circuit is extremely responsive and wouldn't allow my light to turn. My Li-Ion pack runs fine without the protection circuit. But now it is up to me to ensure I don't discharge it to an unsafe level.
I also had to construct my own negative electrode for the pack. I used plastic from a milk carton for the insulator and Solder Wick for the conductor. I know it is horribly crude, but it works.
As for spare Nicad packs, I purchased a 7.2v 1500mah racing pack for RC cars at the local Hobby People store. Once again, I had make the negative electrode using Solder Wick. But I ended up with two more nicad sets, all for only $8.99.