You do have several options:
1- Find high current / fast charge NiCad cells (or pack).
2- Accept short battery life if using high output bulb.
3- If low output bulb is OK for you, Eneloops or other options such as DoR suggested will work.
4- Possible to use 4P/2S IMR 16340
5- Possible to use 3P/2S IMR 16500 (17500)
6- Possible to use 2P/2S 16670 (16650)
7- Possible to use 2P/2S 18650 (if physically enlarged battery case)
8- (My wish is) to use 2P/2S 21700 in redesigned larger battery case. (Two hours HO run time, no memory effect, no self discharge)
When you go option #4, #5, #6, #7 and #8, you should change battery charging system and individually recharge each group of parallel cells. In this case, you should have two female ports on battery case for proper and evenly charging all cells.
If you decided to go #6 or #7, make sure you pick batteries which possible to solder on each terminal. Protected KeepPower cells not possible to solder.
We would like to see which way you prefer or go and how the result is.
If I sent you the light, is this something you could do for me? I have no problem keeping it simple and cost effective, I'd just like to have a usable light.
If so, what's the investment for favored options.