@hevans1944 thanks for all the info, and the solution.
Now I just need to figure out how the LM317T does what it does. I will be sure to follow your advice. And I will check back here when I reach my next roadblock.
Don't follow my advice or the advice of anyone else blindly. Understand the circuit and check the math.
My advice in post #13 was incorrect. The LM317T will not be operating within its specifications if you try to drive an LED with 16.2 V forward voltage at 0.3 A. It will work within specs if the forward drop is only 14.4 V. See below.
You really don't need to know
how the LM317T works. You do need to know its operating parameters and limitations. Treat it as a three-terminal "black box" device that always tries, by some internal (and unknown to you) means, to maintain 1.25 V internal reference voltage between the OUT pin and the ADJ pin, provided there is at least 2 V (but no more than 40 V) between the IN pin and the OUT pin.
The power supply positive terminal is connected to the IN pin. The power supply negative terminal is connected to one side of the load which is the cathode of your LED. The other side of the load, the anode of your LED is connected to the ADJ pin.
Connecting a fixed-value resistor between ADJ and OUT means there will be constant current equal to 1.25/R flowing through that resistor and to whatever load is connected between ADJ and the negative power supply terminal, even if that load is a short-circuit. Especially if that load is a short circuit. There is a maximum resistance that can exist between ADJ and the negative power supply terminal that will allow constant current operation. This is the resistance that causes the compliance voltage to reach it maximum value. This value will be the power supply voltage minus the minimum 2 V IN-to-OUT differential voltage minus the 1.25 V dropped across the current-setting resistor.
For a 19 V supply and 2 V minimum between IN and OUT, the maximum output of the LM317T will be 17 V. Subtract 1.25 V for the drop across the resistor and you obtain a maximum compliance voltage of 17 - 1.25 = 15.75 V. This is less than the 16.2V maximum forward voltage across your LED at 0.3 A, so
the advice I gave in post #13 is incorrect. Bad arithmetic on my part. The circuit may still drive the LED at reduced current if, at that reduced current, the forward voltage of the LED decreases to 15.75 V.
So, what to do about the failure to operate the higher voltage LED within LM317T specs? You could increase the resistor value from 4.166 ohms (a non-standard value) up to 4.7 ohms (a standard value for 5% tolerance resistors) which would reduce the LED current to about 0.266 A. At this reduced current the LED forward voltage drop
might decrease enough to bring its terminal voltage down to 15.75 V. If not, choose a larger resistor until the LED forward-biased voltage is equal to or less than 15.75 V. Or you could raise the power supply voltage above 19 V to restore the necessary 2V differential between IN and OUT. That probably isn't a practical solution for you since it would require another battery pack wired in series with your existing pack. It would also bring the IN-to-OUT voltage dangerously close of the 40 V limit if both packs are set to 19V.
So maybe a fixed resistor between your 19 V battery and the LED is the most simple solution after all. You need to determine whether your LED has 16.2 V or 14.4 V forward voltage at 0.3 A. For the higher voltage, try an ohms value of (19 - 16.2)/ 0.3, which is 9.33 ohms if I calculated correctly. Nearest standard value with 5% tolerance is 9.1 ohms. Power dissipation in 9.1 ohm resistor at 0.3 A is 0.819 watts, so use a 1 watt or greater resistor. It is always okay to substitute a higher wattage resistor for a lower wattage resistor of the same ohmic value. The current will be somewhat greater than 0.3 A because of the lower valued resistor.
If the forward voltage is 14.4 V, the current with a 9.1 ohm resistor will be (19 - 14.4)/9.1 or 0.5 A, which may be too high. In which case, increase the value of the resistor to 16 ohms. The power dissipation in the 16 ohm resistor at 0.3 A is 1.44 watts, so choose a 2 watt or 5 watt resistor. If I were playing with this I would start with the 16 ohm resistor and measure the forward voltage drop to see if I had the lower voltage or the higher voltage LED. If you have the higher voltage LED, the current will be below 0.3 A and you can safely decrease the resistor to 9.1 ohms. I hate this approach because of the large amount of power wasted in the resistor, but there is also power wasted with the LM317T approach.
Please let us know if your LED survived the advice you received here.