Read the datasheet of the LM317, on page 9 it states:
So when you feed the LM317 14 V it can regulate to 11 V and lower, not 13.5 V.
Also there will be 1.25 V across R1 so for 13.5 V you will need to put at least 13.5 + 1.25 + 3 = 17.75 V into the LM317.
The ~15 V you're feeding the LM2596 board isn't even enough, there's no need to have that LM2596 converter in place so remove it.
You will need a power source with a higher voltage than ~ 15 V.
As the LM317 will drop 3 V or more at a significant current, it will get hot so use a heatsink! If the LM317 gets too hot it lowers the current to lower its power dissipation (and allow it to cool down).
Note that your circuit does not have a well defined "stop charging" voltage, current will keep flowing and your battery might over charge!
I have built an LM317 based battery charger for my 12 V car battery. I use a 19 V laptop power supply I had lying around to power it. In that design I do not use the LM317 as a current source, instead I use it as a voltage regulator set to 13.5 V. Then when the battery has a lower voltage, the LM317 will hit its build-in current limit (< 2.2 A). For a car battery 2.2 A or less is fine. As the battery charges an the voltage reaches 13.5 V, the current gets smaller and smaller until only a leakage current is left.
If that 2.2 A is too much for your battery, use this circuit instead: