The supply (3x18650s) were already decoupled at the lipo protection circuit module side (not pictured); I used a combination of 10n and 1µ for each 18650 (with protection module) to be extra safe.
You're intentionally switching 800mA. That part you know about.
This part you might not. Each time the 555 changes output state, it also briefly consumes 200mA. Basically, it shorts the pull-up and pull-down transistors on its output together ("shoot-through"), so this consumption occurs whether you're using the full 200mA rated output or not.
Either of these transients can cause the voltage differential between Vcc and ground to momentarily collapse, if not properly decoupled. The decoupling capacitors should be as close to the point where large currents are used, because all the wires and traces leading up to that point have a tiny bit of resistance and inductance. This becomes significant when working with large currents, even just the 555's own shoot-through; preventing their stored charge from quickly flowing to where it is needed. The decoupling caps on the batteries are too far away to be very effective, connected by relatively long wires. And too small as well.
So what happens when you don't decouple properly, and the Vcc-to-ground voltage collapses? The 555 generates reference voltages for 2/3rds and 1/3rds of Vcc via a simple voltage divider. And then uses a comparator to compare the timing cap voltage against its reference voltages. With Vcc collapsed, the reference voltages change, and it can false trigger, then retrigger almost immediately after Vcc rebounds. Which means it's possible it's consuming 200mA almost continuously. This can be up to 12V*200mA=6W, and you can see why the 555 is heating up.
The wiring on your breadboard version was by luck, just good enough that this didn't happen. Or perhaps you were using a DC power supply with larger internal caps, rather than the batteries. Whatever the case, you need to add that decoupling! Try temporarily connecting the + lead of the 1,000uF capacitor to the + lead of the LED. Also put the largest ceramic cap you have (preferably at least 0.1uF) parallel to that electrolytic cap. Nevermind if it whines, just see if it solves your immediate problem. If so, then you know the only issue is decoupling, and you can work towards a more permanent and elegant solution.