OK, I'll ask a dumb question...what if I want the "brake light failure detection circuit" to work with LED bulbs.
Can you run a resistor in the circuit?
No. This is completely different from the resistor in parallel for turn signals to prevent hyperflash, although it has the same problem in that, with the resistor in parallel, the relay will not hyperflash, even if the LED turn signal is blown, because the resistor is still drawing current.
The brake light failure detection circuit works with 4 reed switches (one for each brake bulb filament) in series. Reed switches are a relay with super low resistance winding that is actually in series with what you're sensing. So for each brake filament, current flowing in that circuit trips the switch, and you need all 4 switches to be tripped, indicating current through all 4 brake filaments, or the warning light comes on. If you wire a resistor in parallel with the LED bulb, that would increase the current and trigger the reed switches, and keep the brake bulb indicator from coming on, which sounds like a win. HOWEVER, if the LED bulb was blown, you'd still get adequate current through the parallel resistor to trigger the reed switch and the warning indicator would not come on. Adding the resistors in series to the 4 bulbs is basically a super complicated way of just grounding the sensor (with 4 resistors instead of simply connecting it to ground) to prevent the bulb-out indicator from illuminating. As noted, LED bulbs last a lot longer, so bypassing the bulb-out indicator is not unreasonable, if you check your lights every season or so. With 4 brake bulbs plus the high-mount, it's unlikely they'd all go unless you had a serious wiring snafu.
In theory, it would be possible to measure the current for each particular LED brake bulb and measure the trigger current on each reed switch and calculate a parallel resistor that would emulate an incadescent bulb, drawing just enough current to trigger the reed switch with the combined current, but not enough with the resistor alone. However, LED bulbs draw such little current that it would need to be super accurate in specifying the resistor and with the resistance changes in a resistor with temperature, it still might not work and you'd need to design the right resistor for every combination of LED bulb and reed switch.