Inductors and caps on buck-boost converter getting very hot under medium-ish load
I've tried some cheap &quot;LM2596&quot; modules and their main problems were: caps' ESR was too high, inductor was saturating, and &quot;LM2596&quot; was counterfeit. So, basically, everything was wrong. Sure your module uses a different chip but considering the price these are usually sold at, I don't see any reason why they wouldn't be the same kind of garbage.
High-ESR capacitors used above their ripple current rating get very hot, which is a problem because they will eventually die, which will cause large output voltage ripple which could harm whatever device it is powering.
Since your router takes 12V input, here's what I would do.
First, check if any parts on the board actually use 12V. You can do this by looking at datasheets, capacitor voltage ratings, or looking at the traces and continuity testing.
If it is a DSL modem/router the analog circuitry could use 12V, I also see a 12V relay on the photo, but these will work fine between 11V-13.8V.
However, most likely there will be a buck converter to step down 12V to whatever voltage the logic chips uses, like 3.3V and some lower voltages for the CPUs and big logic chips.
If this is the case, you don't need to feed it accurate 12V. Just check the buck converter and caps can handle your battery max voltage, and power it directly from the battery with a fuse.