Wire reverse polarity indicator led to handle voltage above breakdown?

I will be implementing the reverse voltage protection circuit shown in this video on a board I'm designing:

I have a red/green led which I will be using for a status indicator, and to reduce unique component count, I figured why not use it for my power indicator LED as well? Then I got to thinking that perhaps I could wire it to not only indicate power, but also indicate reversed polarity so it's easy to tell if the problem is a dead battery, or a reversed battery.

The problem is, my circuit can be powered by anything from 4.5V to 16V, and if I simply placed one led in reverse across the power input, I assume even if I have a resistor on the led sufficient to limit the current to 20mA at the maximum voltage I expect, that I may still damage the LED thanks to overvoltage.

Is that true, and if so, how can I implement this cheaply? I'd rather not add another unique component to the board, like a diode. I was hoping to get this for free.

I considered a voltage divider, but if I divide by 2 while 5v becomes 2.5v which will still be enough to light the led, 16v becomes 8v which is still too high for the reverse breakdown which is 5V.

Why not use a single resistor connected to the two LEDs wired back-to-back with each other. Whichever LED is correctly biased will illuminate and limit the reverse voltage across the other LED to around 2 to 3 volts.

So like this?

That's it? I would have imagined if I measured across the LED that was conducting at best I'd get Vin - Fv, and that's what the reverse voltage across the other LED would be.

Nope, the reverse voltage across the non-conducting LED is Vf of the conducting LED
Vin-Vf is the voltage dropped across the resistor.

Well I stuck a resistor and an LED in a breadboard and measured it at various points and sure enough, 2.5V across the LED, 6V across the resistor, and 8.5V across either end from the battery. I guess this will work then. Thanks!

I have to wonder though why none of my searches pulled up any explanations of how to do this. I actually found one circuit using an opamp to do it, and other places suggested diodes in series with the LED.

Perhaps because 'clever' people always try and make thing complicated.
The golden phrase is KIS Keep It Simple

You were right to worry about reverse voltage, typically LED's have an abs max
reverse voltage rating of 5V (suspiciously they nearly all have the same value,
which probably means its conservative and they don't really measure it!).