I did think of another reason--I had a brief exchange with Jim Bob's sister Deanna on Facebook a few months ago and she mentioned that they have Jewish ancestors--she didn't indicate which side of the family or how far back, but was clear that there were Jews in their family. It may be possible that the Duggars are incorporating those practices in their lifestyle as a homage to those ancestors. I have no idea if it's a reason, but it's again, another possible theory.
As for "picking and choosing" ...well I think pretty much everyone does that, whether they want to admit it or not, or whether they consciously mean to or not. Jews do it--no two Jews are exactly alike (ok Chassidic Jews certainly try harder) but there are different movements within Judaism--Reform, Conservative, Orthodox, Chassidic....and different beliefs within those movements. There are even more different subsections of Christianity and all have different methods of practice--who's more Christian than the other?--there's an argument for you...
I think the whole point of faith is that you practice what makes you comfortable--that's what faith is--it's what gets you safely through the day without fear or panic--knowing G-d is taking care of you--believing G-d is taking care of you. So you go about that in whatever way brings you peace. Some practices may not bring peace to OTHERS, but that's really OTHERS concern. Some may see other's practicing as incomprehensible, but you need to practice in a way that makes you comfortable. For example--the Christian church run by Fred what'shisname that condemns homosexuals and pickets funerals--i find that offensive, but that's what Christianity is to them and what makes them feel like Christians. I worship at a Chassidic temple, but I'm quite sure that Chassidic life is not the Judaism I was personally meant to practice--it's not quite where my personal beliefs lay--but I certainly support my Chassidic friends in their quest for faith, as they support me in my quest for mine.
I think the Duggars call themselves Independant Baptists for a reason. They are trying to forge their own path with G-d's guidance. A lot of what they do doesn't mesh with mainstream Christianity, but that doesn't make them any more or less Christian. It makes them faithful and makes them pioneers in a sense. All branches of Christianity started out with pioneers.
And one more thing, it's my understanding that Christians aren't supposed to judge others. I think calling them out for their methods of religous practice is a bit judgemental and unfair.