I've thought of an excercise that can knock 2 birds with 1 stone. You can walk away performing a spiritual demand by the Watchtower, while also introducing TTATT to your family.
I did this once with my roommate way before I knew the TTATT, we wrote questions on the back of flash cards, choosing them randomly to create realistic situations that you may encounter at the door. We would role play between being the house holder and the JW. This excercise helped prepare us for difficult questions to keep us on our feet. If we didn't know the question we would research it and prepare a response for the following week. We purposely created situations to try and stump each other, or see how well we did under live pressure, it was a lot of fun and turned into a game of wits.
Now that I know the TTATT, I have realized this excercise can be of great tool to help others wake up. I would introduce this excercise as a way to help develop spiritual knowledge and sharpness at the door. You should first prepare some pretty easy stuff and do your best to establish this as THE FAMILY WORSHIP activity of the week. Make it fun and challenging. Stay clear from apostate questions or areas out of their intellectual realm for the first week or 2 (dumb it down if you have to). Display a very Pro JW attitude if necessary to destroy any apostate fears/suspicions (fake it till you make it). Progressively start asking difficult questions which will require them to research. For instance, the first flash card could have something like this:
House Holder: "I believe Jesus is God Almighty, you Jehovas need to believe in the Trinity!"
JW Challenge: How would you answer this objection using only the bible?
Of course this is easy stuff that can be defended by many parroted Reasoning book answers. Let them answer these without debate and sincerely feed their confidence in the things they do know how to defend properly. Now when the time comes, bring out the fireworks. Progressively start scattering TTATT questions in between the easy ones. Below are a few examples...
House Holder:
"Can you explain the new Faithful and Discreet Slave light?"
"Can you explain the overlapping generations teaching?" --Follow up question, why has your definition changed so much over the years?
"What's the context of Proverbs 4:18"
"Can you show me how you arrive to Christs presence in 1914 from the bible?" --Follow up question, "Why did Witnesses believe Christ arrived in 1874 then?"
"Why do museums, historians, and archaeology state Jerusalem fell in 586/7 instead of 607 bc?"
"Based on what evidence or beliefs were JW's chosen to be the Faithful Slave in 1919?" --The Finished Mystery will be very valuable here.
"My preacher tells me you guys are false prophets!"
"Why did you guys claim the end of the world would occur in 1914, 1918, 1925, 1975?"
"Why do you guys not celebrate holidays but still oberve pagan traditions and rituals?" -- wedding ceremonies, wedding rings, jewelry, anniversaries, days of the calendar, braiding hair, make-up, etc.
"Why were organ transplants considered cannibalism in your publications?"
These questions need to have the premise and challenge of being answered using only THE BIBLE. The use of publications is alright but again, make your family understand that Watchtowers are not great proof at the door as the house holder will only see them as JW interpretation. Throughout this excercise you should try to remain very pro JW as to not give away the motive. Make them know you're in the position of playing devil's advocate throughout these challenges. Do not be a know it all, or smart ass during these as they will not take you seriously. Encourage an environment of love, openness, and free speech. This hopefully will get them to think and wake up when they have the burden of proof placed on their shoulders.
EDIT: I forgot to mention that if you manage to develop a JW library, this can be SO valuable. Having the evidence and historical references right there available for your family to research the challenges. Imagine creating an assignment for the family to inspect the very food Jesus himself inspected to anoint the Watchtower as his Faithful Slave! The greatest honor in all human history! Create projects or assignments that will require them to read: The Finish Mystery, Studies in the Scriptures, Millions Now Living Will Never Die! etc. Eventually you can bring in friends from the hall into these family worship nights too if you want to work on persons who are not relatives.