Trial and error....
I'll paraphrase...
When the brakes are NOT on, that
sense line is 0V (GND) since the brake bulb filament is a
short to GND - ie, it connects the line to GND.
It's an '
absence of GND' that tells the
module that the brakes are on. (Hence disengaging cruise controls, and apparently your problem.)
LEDs mean an absence of GND due to their voltage drop.
As to what resistor is needed...
We know that a ~4 Ohm or 2 parallel 7 Ohm resistors will work since that's the equivalent of 2 stop bulbs. [ R = VV/P = (12V x 12V)/(2 x 21W) ]
One 7 Ohm should work unless module designers thought it was a 'feature' to have the module disengage if one stop light blows.
Problem is that we need 7R (Ohms) at ~20W which is big, expensive, and a lot of heat (ie, about the same as the original bulb which is what we are imitating).
In practice it's more likely that the module can work with a far higher resistance.
Internally it will have some
pull up resistance to keep it at +12V. The stop filament then pulls that low (ie, to GND or 0V or near 0V eg 0.7V or 12V or 2V etc).
Such pull-up resistor are likely to be 1k or higher (since we want to limit resistor ratings/sizes to typical 14W or 1/2W types).
They might even be 10k or higher.
It's a voltage divider...
It's a
voltage divider comprising the module's pull-up resistor and the filament resistance. (eg 1k or 10k etc with 7R or 14R etc.)
Now, assuming it's 1k and it needs to be pulled lower than (say) 0.7V, that means a (clink zap boom kerchunk...)
47R 5W resistor.
But maybe a higher resistance will work.
If you have resistors on hand, give them a try. IMO you want as high a resistance as possible (only for less heat and hence a smaller sized/Wattage resistor), but it must be low enough to ensure proper operation.
Note that the
minimum value resistance that can be used
depends on its Wattage - eg:
1/4W 820R
1/2W 390R
1W 220R
2W 120R
5W 47W
- where 'R' means Ohms (Ω)
... and the above are based on preferred resistor values and assume a voltage of 14.4V (the resistor is across the +12V supply when the brake switch is closed).
Test first using bulbs?....
Keep in mind that you can try first with a stop bulb (21W => ~7R), a tail bulb (7W?), dash bulb (2-3W).
If a 2W dash bulb works you know a 120R 2W resistor should work.
A 3W bulb is between 56R to 68R, but that would mean (say) a 56R 5W else 2 parallel 120R 2W resistors.
Sorry if that's too much verbiage but I've included assumptions, justifications, and various test & permanent methods & sizes/ratings. Even then, a lot of detail/theory has not been given.
Please ask if you want a better breakdown (or just treat each section in isolation). Otherwise maybe it narrow down by saying what you have available, or what any testing shows (eg, it works for a 7W tail bulb not a 680 Ohm 1/2W resistor).