Further tests show that the incredible little A2SHB MOSFET wasn't actually getting flustered at all. The heat dissipation from the adjacent resistors were the real culprit.
I did remove one of the 0.5 ohm resistors and it had very little effect on the LED current. It dropped from 1100mA to 900mA suggesting that other components, tracks and wiring are being pushed too hard. You might consider swapping in a new resistor to choose a current more suited to your application, balancing between run time and intensity. An example might be to use around 2.2 ohm to drop the peak current to around half an amp or ten ohm for much longer winter run time.
The lithium cell capacity is a reasonable 1200mAh, which was better than expected, and quite acceptable for this type of product. You could upgrade the cell to a larger capacity, especially if you added a larger external solar cell with an output of 5V or more at up to 500mA. That could work well in places with low winter sunshine.
Here's a generic search linkon eBay to find this type of light:-
Target price is around $8
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