Nice and simple mod, but it does come at a price. Those 12V batteries have very little power in them and 3/4 of that gets wasted in the resistor.
Instead of a resistor you can use a single 7135 chip to maintain 350 mA output.
That battery only has 55 mAh of capacity. Plus 12V will kill a 7135…
Pardon, it does work fine with a 10440 lithium.
Nicely Done! I really like the resistor idea.... Good THinking!!
Dan.