Will this LM317 Circuit work for my application?
I agree with @mkeith's comment to monitor the voltage (relative to ground) at the LM317's power input pin when the fan is attached to the LM317's output pin, and the fan is running. If at all possible use an oscilloscope, not a multimeter, to perform this voltage measurement.  (NB: If you don't have an oscilloscope, use a multimeter to measure this voltage.)  If the measured voltage at the LM317's input is at or below

$$
11.5\,\mathrm{V} + V_{DO} \approx 11.5\,\mathrm{V} + 2\,\mathrm{V} = 13.5\,\mathrm{V}
$$

where \$V_{DO}\$ is the LM317's dropout voltage specification (obtained from the LM317's datasheet), then you know the 19&nbsp;VDC power source cannot deliver enough power (voltage and current) to the load.  The only option at this point is to replace the existing 19&nbsp;VDC power source with a different power source that can supply enough power (voltage and current) to operate both the LM317 and the load.

(P.S. Be sure to have a fire extinguisher handy and always wear eye protection if you're going to try charging lithium batteries without a BMS. See also this YouTube video.)