Well, I'm interested by the question and waiting for some correct answer from high levels, but seems no one answered... So I'll write down what I know and guess, and hope that the more detailed and correct answer will show up later...
For most opposite reputations, raising one will reducing another. So if you want to get both badges, you should push one to the top first, then work on another.
But for BoV and BoE reps, according to the library, raising BoE rep will reduce BoV rep, but raising BoV rep WON'T reduce BoE rep. (I am not sure if it's true or the library missed that point, but it's very likely true, I'll tell reason afterwards.)
If so, what you need to do is to get BoE red badge first, then BoV red badge. (But remember only BoV red badge needs totally around 1450 EEs, BoE needs more...)
Now, I'll tell why I think you can really get 3000 rep for both BoV and BoE. There are some items, their function is to let you switch between BoV and BoE. They are
Versatility Charm and
Scroll of Separation . Check the descriptions and you'll know, you can use Scroll of Separation to get
Half an Evil Eye , with 30 Half an Evil Eyes you can recharge the Versatility Charm and use it to switch between BoV and BoE once every 10 days.
BTW, following is the requirement for joining BoV and BoE. You need to be lv5 to join them. For BoV, you need to be a lv2 healer or lv30 locksmith. For BoE, no other requirement. And both initiation quests are not complicated, so I don't give details here.