Did someone say LM3914?!
I, too, was very puzzled when I first played around with this chip. Like the veritable 555, once I looked at the block diagram, things fell into place. It also clued me into how to adapt this thing for my nefarious purposes.
The first thing that'll help you is to view the 3914 as three (or four, if you want to include the MODE operation) sections: some LM339s, an LM334, and a driver output (like the ULN2003). It makes sense, as all of these are National Semi designs - they just shoved all of it onto one substrate.
Section 1: Comparators (LM339s)
All we have here is a simple ten-segment resistor divider ladder between one input of each of the LM339s, and the reference (buffered) from the SigIn tied to the others. This gives us, in the case of the LM3914, a ten-step linear divider, using 1K per step, for a total of 10K (<--remember that!). In fact, you can build one from 3 LM339s, but it'll take up more space. The LM3915 has deciBel stepping (about 22K total), and the 3916 has VU steps (~10K).
Section 2: Voltage Reference (LM334)
Well, if we want to measure against a stable reference, why not shove one of those in there, too? The nice thing is they didn't connect the LM334 section (REF Out/Adj) to the ladder (Rhi/Rlo) internally (^--remember that, too!).
Note that the LM334 provides for the LED driver current too, so it serves double-duty in this chip.
Section 3: Driver Output
Simple 2N2222-type open collector outputs, with current limiting provided by the LM334. One resistor sets the current for the whole string of LEDs, and is why the LEDs don't need any dropping resistors.
Operation: The goofy game for eccentric EEs!
Now, the problem I had, and I suspect you're having, is that pesky REF Out/Adj. All the circuits in the datasheet all seem to use it, but playing around with it doesn't give you the desired results (or range of values you need). This is because it's only 1.25V nominal. If you feed the REF Adj pin about 4.3V, then tie the REF Out to Rhi, that'll give you a 4.3-5.2V scale. Great for TTL logic, but sucks for car batteries, huh?
It was after looking at the block diagram for the umpteenth time that the epiphany hit me: Fsck the REF! Just tie the ladder (Rhi/Rlo) between a couple of pots, give it some stable Vcc, and you can set the ladder where you need it, and even adjust for how much each step takes. The folks that designed this thing probably thought that, too, and is why they brought out the pins instead of tying them internally. Brilliant! So, all you need to do is take that 10K ladder, offset it from ground to get the desired Lo turn-on value (Ohm's Law), then set the High value based on whatever value your Vcc is. You'll see this in the attached schematic I whipped up. It scales perfectly, too, as all you need to do is wedge in the additional 10K ladders of however many LM3914s you want to use.
Problem is, we can't just forget about the LM334 section entirely, as it provides the LED current. So, tie the REF Adj to ground, which makes the REF Out 1.25V. Throw a suitable resistor from REF Out to Ground, and the current pulled will set the LED current thus: ((1.25/Rled)x10)=Iled. So, a 1.25K resistor will limit LED current to 10mA. With modern efficient LEDs, I just use a 1.8K (~7mA).
We now come to how to measure the voltage of a device that powers the tester we use to measure it. Your original schematic had this part right: just divide the measurement voltage with a simple resistor divider. The LM3914 SigIn only needs a few uAmps, so current draw won't be a problem. So, we use a Vreg that'll give us more than half the expected low battery voltage, and tie that to our Vcc and modified ladder. If voltage seems to dip too much for our test, then we can just use a divide-by-three (or four) on SigIn. Another beauty to this is that the whole shebang (using Bargraph Mode) will draw about 50-400mA, which gives us a pretty good load to put on the battery. If you wire this up for Dot Mode, each chip will draw about 10mA, and allow 20mA for the LEDs (the LM3914 will keep the previous LED on when switching to the next, so current draw can be two LED's worth). So, using 3 LM3914s in Dot Mode should draw about 50mA. Still not too bad.
Hope all this helps you out. I can't test this circuit, as I only have one LM3914 left, but the single-mode version I made works well (with a 555 driving it).
nop
TL;DR - Offset and daisy-chain comparators - skip the REFs, but set the current - use Vreg - measure half Vin.