Why is my low dropout voltage regulator Diodes AP7370 burning up?
If the LED is drawing its power through the 7805, the regulator will require a large heatsink.  If the bench supply has an adjustable current limit restricting the output to a max of 1 A, this might be why the regulator survives bench operation.  A lithium battery has no such limiting.
BTW, the 7805 is not a low-dropout regulator.  It is a second-generation single-chip regulator from the 60's/early 70's, and needs over two volts differential between its input and output to work properly.