5s LiFePO4 cells with 2s 6V XHP50 might work.
I put together a tool for predicting current in different direct drive situations that might help you here. Basically you graph the supply voltage as a function of current which includes the voltage drop from all the different resistances in the circuit. Then you graph the forward voltage of the emitter. The point where the two curves intersect is the current that will result in a direct drive situation. I explain it more here: Tool/method to predict current in direct drive setups
Put this in google: graph 3.27*5-x**(.015*5.01.01) and 2**(.136*(x/1)+6.236)
x in these functions is the current in amps. The 3.27V is approximately the resting voltage of LiFePO4 cells, multiplied by 5. The next term is the voltage drop from all the circuit resistances. I approximate the IR of a cell as 0.015 ohms, times 5 for the 5 cells in series. Then the .01 + .01 are other resistances like FET and tailcap and wires, etc.
The other function is the forward voltage of the XHP50, which I approximated as a line from data from djozz’s measurements. The factor of 2 at the front is for two 6V XHP50s in series. The (x/1) means there is one parallel string, if you had 7 XHP50s in parallel you would put in (x/7) here.
You can get a good idea of what setups will work with this tool. To use Li ion instead of LiFePO4, put in 4.2V (or the voltage at whatever state of charge you want) and a suitable cell IR.
Hope this helps and please ask any questions if you have them.