Low-power laptop

You are not going to do much better than what you already own. First you should forget about your power bill unless you are a serious jew, at peak power a 35W CPU like typical in laptops from around the X200 times will cost you like $2 a month.

For battery in emergency, an X200, which you already own, and which I also own, lasts around 5 hours on max brightness watching Chinese cartoons with a fresh 9-cell battery when I have used it. If the screen is off and you're not doing anything I wouldn't say it's too farfetched for it to be 9 or more hours. You can extend that by getting an Ultrabase and the Ultrabay battery (Battery 42) which gives you probably %25 more juice.

If that's too much to deal with then yeah just get a shitty Atom netbook. I guess the X131e might be alright even if you can get a decent healthy battery with it.

just plug a screen and a full keyboard into the RPi or other SBC
problem solved

the official rasberry pi screen is $75 I guess, but 5 minutes of searching on amazon; amazon is filled with chinkshit that comes with a screen and a case for ~$30

I don't think it's getting any lower power or cheaper than that.

get a car battery and an inverter or adapter to use the DC directly

You can plug a car battery directly into the charging port of your laptop. Make SURE to get your polarity right or it will blow up completely, and don't short anything out. It will run off the car battery, but won't be able to charge the laptop battery.
THIS ONLY works if your normal charger is 13-22V. Anything lower will blow up and anything higher might not work at all.

i wouldn't trust that at all; it would be an interesting project but I would want a fuse or something in between and controlling the voltage.

easy mode would just be use an inverter and plug it in like anything else but the conversion would waste a bunch of power

An used netbook. I'm pretty sure there's tons of e-waste netbooks that are still perfectly usable.

Alternatively, an android tablet with a cheap OTG or bluetooth keyboard. What do you need this for?

this sounds wrong, why
it should be
"A used"

and that just tells you how the each writer pronounces "historical".

Netbooks?
some eeePCs are around 10W, comparable with some power hungry SBCs
silentpcreview.com/forums/viewtopic.php?t=51102

Alternatively get a Raspberry pi laptop / laptop kit.

eeePC = Atom CPU