This scheme creates a new "phantom load" on the batteries at 3 watts (about 1/4 amp). You need to keep the PID powered from the batteries at all times, in order to make it keep its settings when you're not plugged in. That also means that you can't leave the batteries in the TM all Winter, without plugging them in at least a couple of times.
Within the containing box of a home-built LPF battery, or surrounding the battery case (for a pre-built "battery pack"), you will want to add a have a low-powered 12v warming pad. But, you don't use a warming pad which contains an RV "freeze protecting" thermostat. Rather, something like this:
https://www.amazon.com/dp/B084YXXM2P/ You also need to buy and program an external Temp Controller, in order to warm the battery pack to at least above 40 degrees (Fahrenheit) before charging it with a high rate of current.
A 5-pin Automotive Relay is switched by the On/Off Temp Controller Relay. At less than 40 degrees, Power Converter output goes into the Relay "Common" (with about +14.5 volts) gets sent out on the "switched" leg into a heating pad. When the Temp Controller turns off, at battery pack temperature of around 50 degrees, the Relay NC leg (closed when the Coil is NOT pulled in) becomes active. The same 14.5V Power Converter Input is now connected to charge battery string. You cannot use an automotive Relay BASE), because the charging current is too much for those socket bases (even so-called "high-current" ceramic ones). But these Relays do the job, with much bigger "power" pins for you to connect 9.5mm female push-on sockets without a base:
https://www.amazon.com/Ehdis-JD2912-...dp/B01IX7NV0C/
In a cold campground with plugins, it works exactly like plugging within a cold garage - the heater will run as needed, and then the Power Converter will run until it is "satisfied". The heater will begin to run whenever the battery temperature falls "too low", which I will consider to be about 42 degrees, and stop running when it hits 50 degrees.
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Solar Power, unfortunately, is more complicated. The battery must be connected to the Solar Power Controller at all times. A switch can be added on the Solar "PV +" input connection (or a circuit breaker can be switched off), to be left off during a cold night. A separate power wire from the battery could run through another switch to drive the battery heater directly from the battery. I will be using a bigger wire (#10 AWG) for the first "coil power" wire, and tap into that for the Direct-To-Heater switched link.
For me, those two solar-related switches will be a manual process. I can open up the "PV +" Circuit Breaker at bedtime, but there is no point in keeping the battery warm until after a while after sunrise has arrived, and the effective charging day begins.
The Temp controller is this one. It is not a PID, it is a simple On/Off switch for heating (or cooling, which we won't be using):
https://www.amazon.com/bayite-Fahren...dp/B011VGAPOC/