Well if you don't have any resistors laying around get some tomorrow, a few packs from Rat Shack will do, get the 100 Ohm ones for this experiment...
I stole DrewSmith007's circuit, see attached...
This is a test circuit because predicting how the LEDs will dim is guess work... And this test circuit is setup to use all 100 Ohm resistors to help narrow down what values you actually need, this allows you to purchase one value resistor and then experiment, keeping it simple...
So using 100 Ohm resistors, wire it up like shown in picture #1, see what happens, this is for a 6 volt battery source... If the white LEDs are still too bright, one at a time remove a resistor on the White LED side, the standard Red LED should be happy as shown... If you get down to one resistor on the white LED and it's still too bright, then refer to picture #2, and repeat the process of removing one resistor at a time, except this time, remove them one at a time from staring at column 1... Example remove a resistor from column one, then remove another from column one, until there is only one left, then start on column 2... Not the most 'efficient' method to figure this out but a straight forward one...
If all works out you will find the sweet spot then, it's just a matter of working a little simple match to see what the final resistance is...
I know "what a hassle" but unfortunately you are the only one that can see the relative brightness of the LEDs and short of using a pot or variable resistor and measuring it after the fact this is the most straight forward way to narrow it down...