It's not a "double dip" for benefits under US code unless both programs pay tuition (you can't pay for the same classes with 2 federal tuition funding methods-which is why STA-21 OCs can only use the GI Bill to cover courses not already covered by the tuition stipend). Since BDCP only pays the member and does not pay tuition, if the GI Bill in any form was previously earned it can be drawn on at the active duty rate.
Pretty sure this is true. As I understand it, the post-9/11 GI Bill pays E5 BAH, tuition (up to $20k), and a stipend for books. The old GI Bill just pays cash.
While in BDCP, you are are considered "active duty" and will receive BAH commensurate with your pay-grade (E3 -> E5 normally. Not sure how they handle prior enlisted of higher rank?). In addition, many state and local colleges will give you a discount on your tuition if you are active duty.
Are BDCP folks are still eligible for tuition assistance? If so, the discount your school may give you and TAS will probably wipe out state tuition. Using your post-9/11 GI Bill may only net you a yearly stipend and an increase in BAH. If this is the case, you just have to decide whether it is worth it to you, or if you want to save it for grad school, a spouse, maybe children?
I believe the older GI Bill should still pay you your full disbursement in cash while in BDCP. In that sense, yes, you would be able to "stack" them.
Worth thinking about.
*Note- I'm not in the Navy. Just a guy who's read up on this, looking to apply to BDCP soon. Certain forum members should be far more knowledgeable on this topic than myself. If I'm wrong in any particular detail, please feel free to correct me.