I took an introductory class for color theory and learned that if you have constant exposure to a certain color, your eyes will crave its compliment. The red appears to have an orange undertone. Orange and blue are complimentary colors so if you buy say, the blue first, you will likely want the red next (or vice versa).
You would probably wind up getting the etoupe last if you were to ultimately buy all of these Birkins.
If you want your bag to go with your wardrobe, I would say a neutral would be the most practical. I would get an etoupe, black or gold first. Those are classic colors and will go with anything. Then I would get the different colors later. Good luck!