If you charge your little batteries at 2.4A (the typical output current of a PB137) then they might be destroyed.
With a 19V supply it will dissipate [(19V - 13.7V) x 2.4A=] 12.7w and will need a pretty big heastsink.
But if your batteries are old and discharged then they will not charge.
Sorry, the PB137 and the LM317 both have a typical output current of 2.2A, not 2.4A. I read the graph wrongly in the datasheet.
Their minimum guaranteed output is 1.5A.
The max current of some LM317's is 3.4A. That will blow up your little batteries.
The project I'm working on requires a 7AH battery in long term service outside, summer and winter. Considering the temperature swing I figure temp comp is important so I don't cook the battery (or worse, cause it to vent) on a 100 degree day. I looked at all kinds of schematics from simple to exotic but once I found the silvertel unit I quit looking.