I've been doing a fair bit of late with some battery-powered LED applications, a bit like you Woof.
I've come to the conclusion that series linear regulation is a dead-end.
It does have the benefit of simplicity, I'll grant it that. But that's about its only benefit.
The wasted heat is power your battery is giving up that SHOULD have been used in making light.
So either you get less light output for you battery, or less run-time.
I've been making some indicators. These are 4 x 1W LEDs, I believe very similar to the ones you are using.
The PCB acts as a bit of a heatsink, but I'm also not driving them to their full power, to reduce heat and extend life.
The HUGE difference here is that with this circuit, I can run 4 or more white LEDs (or any other colour!) at high brightness, off a SINGLE 18650 CELL. (It'll work from about 1.8V to 5.5V). The boost converter uses a very small value resistor (I'm using 2.2 ohms) to set the current, so there is almost no heat or waste power in the resistor. Under a thermal camera, the only part of this board that gets detectably warm is the inductor, and even that is only about 4 degrees above ambient.
In the interests of power conservation, I would urge you (and anyone else dabbling with power LEDs) to skip the series resistor, and/or linear regulator solutions, and go straight to a switch-mode converter using a very low voltage sense (feedback) control. The one I'm using tries to maintain 0.25V across the feedback resistor.
Additional benefits they bring to the table is a "brightness" input pin that can be driven by PWM, or by an analog input, to dim the LED from basically nothing to 100%, with virtually NO LOST POWER. Nothing gets warm. 92% or 96% or something of your battery power gets to the LED, none of this "dropping lots of volts (at full LED current)" nonsense.
Just to demonstrate the point - in your above linear option, you're going to drop 1.25V across your resistor 100% of the time. So lets work at 100mA (where I have mine set). If you have 4 LEDs at 2.3Vf that's 9.2V across the LEDs. With your battery full and sitting at say, 12.8V, that's 3.6V you have to drop across the regulator. 1.25V across the resistor (125mW) and 235mW across the LM317, for a total of 0.36W wasted as heat from your total of 1.28W drawn from the battery.
As the voltage goes up, the worse this gets. At 13.8V, it's 4.6V to drop, 460mW wasted from 1.38W drawn from the battery.
The converter option with 0.25V sense would be about 0.99W drawn from the battery with 0.07W wasted as heat for exactly the same LED output!