I think the RC circuit is the right solution to this problem.

What you need to do is to add an electrolytic capacitor across the terminals
of the LED.  This will cause a slight delay when the circuit is energized
and will eliminate the flashing if the capacitor is large enough.  If you
know someone with a scope you could find out what the actual pulse width is
but failing that I think that a reasonable delay to try would be 10ms and
normally this would required a capacitor value of  C=T/R or .010 seconds
divided by the ballast resistor value which might get you a capacitor spec
of 500 microfarad.  Unfortunately, the LED will light up when the capacitor
voltage gets up to .6 volts so we need to increase the capacitor spec by a
factor of 15 to get the 10ms delay.  This would require a 7500 mfd capacitor
at 25 volts working voltage.  These are pretty bulky and hard to come by so
I think I would try a 4700mfd at 25 volts  as a first attempt.  These are
about 3/4in in diameter and 1 1/2 in long so there's some hope that six of
them might fit in your housing.  I think you should be able to get them at
Radio Shack or failing that at Digi-Key for sure.  Don't skimp on the
working voltage.  Be very sure that the positive terminal of the
electrolytic capacitor is connected to the positive side of the LED.  If you
hook it up backward, the capacitor will short out.  Because we don't know
the actual value of the ballast resister and the required  capacitor value
is uncertain, I would just try it on one LED at first to see if it works.
If the test LED still flashes put another capacitor in parallel with the
first one.  If you know the actual value of the ballast resister let us know
and we can be more specific with the fix.

Good Luck

The LED may be low resistance, but couple it across a 12 volt supply and
it will die instantly. So you need a current limiting resistor wired in
series. For car use and most single LEDs this might be around 330 ohms.

A bulb with a resistance of 240 ohms would be approx 0.6 watt at 12 volts.

The centre brake light on my E39 is 21 watts, so the resistance when on
will be approx 7 ohms - less when cold.

I've got Hella aftermarket LED taillights on my E39, and there are two
rather large units which fit under the spare wheel to compensate - for the
warning device. I dunno how they work, but suspect there's a bit more to
them than just a resistor.

Been there done that - I teach electronics to 11 - 16 year olds and have thought
of most things.  21W bulb has a current draw of about 1.7A and about 7 ohms
resistance.  6 x 0.02A = 0.12A current so the resistance is around 100 ohms in
theory.

Taking theory out of the equation the LEDs actually short circuit the system and
light up because of the 100 ohm resistance or thereabouts and the incandescent
bulbs just let the current pass.

I am not shutting you down just putting the theory to the test.  I have stuffed
a 10 ohm resistor in series and even tried it in parallel but still it flashes