First things first I'm an EE so let me rant about units ;)
I have modest power requirements but do need to top up a 2350 mAh headtorch battery every day and a smartphone every 2nd or 3rd day.
The unit of energy is the Watt*hour (Wh).
The unit of capacity of a battery is the Amp*hour (Ah) which is 1000mAh.
If a battery has a capacity of 1000mAh (1Ah) and a nominal voltage of 3.6V then it can output 1A*3.6V=3.6W for 1 hour, so its energy capacity is 3.6Wh.
If a battery has a capacity of 1000mAh (1Ah) and a nominal voltage of 12V then it can output 1A*12V=12W for 1 hour, so its energy capacity is 12Wh.
I need to bring all power and I need approx 200.000mAh/200Ah of power.
Now mAh isn't a unit of power, and since you're mixing all the units, I'm gonna bet your numbers are probably wrong too, which could be a problem ;)
I have modest power requirements but do need to top up a 2350 mAh headtorch battery every day
Assuming the usual LiIon, 3.6V*2.35Ah = about 8.5Wh/day for the light, so you'd be running at 0.85W for 10 hours, sounds legit.
and a smartphone every 2nd or 3rd day.
Is there cell coverage where you're going? Unless you intend to use it to listen to music, then alright. Say 4Ah, 14Wh every 3 days, 4.66W/day.
So that's 13 Wh/day, let's round it up to 25 just in case. 30 days, 750 Wh.
750 Wh is 62.5Ah on 12V. So I guess you forgot the voltage when going from Ah to Wh so your planned 12V battery is about 12/3.6 times too big!
GPS unit etc. simply run on ordinary lithium AA batteries).
OK.
Anyway. Let's list the specific energy of various batteries...
Non rechargeable:
- Alkaline 65-100 Wh/kg @ 1.5V, 0°C/60°C
- LiFeS2 300Wh @ 1.5V, 0°C/60°C
- LiMnO2 (expensive) 280 Wh/kg ] 3.3V, -30/+60°C
Lithium rechargeable 100-265 Wh/kg, which depends mostly on packaging. For example a pouch battery does not come in a heavy metal tube like a 18650 so it is lighter, but it has no protection, so if you puncture it, it will probably explode. If that happens when you fall on it because you were carrying it inside your clothing to keep it warm, you may have a bad day.
Every inefficiency and energy loss you incur increases the amount of batteries you have to carry. So if you charge your headlamp from a powerbank, assuming 90% efficient conversion from 3.6V to 5V in the powerbank, and 70% efficient cheap linear charger, you're at 67% efficiency, so you're wasting more than 30% of your battery weight.
I like redundancy so I'd use a LED headlamp that takes an external battery. Here's an example. Since the standard battery format is 18650 this means either a headlamp with an external case where you can stuff some 18650 LiIon cells, or something like the one I linked which plugs directly into a USB power bank.
Headlamp+powerbank: you eliminate the worst inefficiency which is charging, so you go from >30% losses to 5-10% losses in the powerbank.
Headlamp+18650: optimum efficiency, since there is no conversion. You grab a fresh 18650 and stick it in.
The 18650 plan also works with LiMnO2 non-rechargeable 18650 cells which are damn expensive but still work at -30°C as a backup plan just in case.
The other issue with Lithium is temperature. At cold temperatures, they lose a lot of power output. They don't lose their stored energy though: a cold battery will appear to be discharged, but it is simply too cold to work, doesn't have any power, so the device thinks it's discharged. But the energy is still there, and if you warm it up, it will be able to work normally.
Charging a frozen lithium battery is not allowed. At these low temperatures, the ion mobility is too low so the lithium ions can't go where they should fast enough, and instead we get lithium metal dendrites forming which short the battery internally. It will then either discharge quickly (ie, it is screwed) or catch fire. In both cases, it's not reversible and the battery is kaput.
Lithium batteries with internal heaters use their stored energy to heat themselves, so that's another inefficiency.
The external headlamp battery can go to a warm place, like inside your clothes, where it will stay warm.
What I'm suggesting is either a bucket of pre-charged powerbanks, or a bucket of pre-charged 18650 batteries (in their isolated plastic cases), plus two headlamps compatible with the choice of batteries. In the 18650 case you also need a converter to get 5V but that's readily available.
In both cases you have a lot of redundancy.
It's not a problem if you store the batteries in the cold as long as you heat the one you're going to use before using it. For example you can keep it in your tent.
It would be a good idea to get a vacuum sealer (very useful for sealing food) and to seal powerbanks. This will avoid condensation inside as you warm them up. Of course you have to break the seal in order to use it, but that's after warming it up, so the worst of the condensation has been avoided.
I do not like the big lithium car battery because you won't be able to put it in your jacket, so it will be frozen. It is a single point of failure, non redundant. On the other hand being intended for automobile use it is much more likely to be designed to work in all weather conditions, compared to a powerbank which is more of a temperate indoors device.