Nosey questions... who lives in your home? Is your nephew living in the house?
I agree with the comments that if your 6 year old nephew is running the show and telling the adults what gonna happen, a parrot is going to terrorize the home. Its big difference between tiels and a caique. Caiques are funny, outgoing and jokingly referred to as the bipolar parrots in the bird world. Your nephew is bullying the bird and while I can't tell you what to do with your nephew as duct tape, rope and a gag in the closet is probably still illegal. You have not mentioned the parents in this situation, while having a family bird is wonderful, who wants to be part of a family where you get mistreated and terrorized. While I understand this is not your child, a serious talk with the parents about what will happen to your nephew's flesh might be needed. While a caique is a smaller bird, please do not underestimate their ability to do some serious damage to flesh, while they may not send you to the hospital needing stitches, I swear I'll take a pressure bite from my macaw over a grinding slicing bite from a conure, sennie, caique size bird. Just personally speaking, if the kid does not live in the home, the bird does back to the safety of his cage when the kid is visiting. If he lives in the home, then guess its time someone set boundaries for the kid even more important then boundaries for the bird.
The dogs... the answer is simple... you never introduce dogs and birds. I know there are those quite videos you see out there, but you already know you have a dog with a high prey drive. You essentially are bringing home a new play toy. The best thing is to keep them separate. When your bird is out with you and a parrot needs to spend time with their flock, being kept in a separate room all the time is much like solitary confinement for people, its not healthy .... parrots belong in a flock, and you are their flock. You definitely need to determine a schedule where the dogs have interaction and then the bird has his safe interaction without the presence or threat of the dog. Birds will fly, and keep in mind that dog with a hunting instinct will catch the bird mid flight or before it lands on the ground.
Caging... the best thing for the bird is to have two cages. A cage in the general family area where he can safely be in the center of activity in a cage if the dogs are in the room. And a cage in a spare bedroom where he can get quiet, secure and safe sleep for 10+ hours each night, and a safe cage with the bedroom door shut to keep the dogs from knocking over the cage when you are not home.
Wish you the best of luck, but really serious attention to safety needs to be considered. Sometimes animals just don't mix... I personally have a macaw that no one told is supposed to be up high, he plays on the floor, and does not like play stands, he has tunnels he plays in and toy several toy buckets around the home. As much as I would love to have a dog, I know for safety purposes, I cannot have a dog in the home without totally changing Max's lifestyle, and that's not fair to him. Not suggesting anything, just sharing the decision we've had to make in our home.
Do wish you the best of luck with the challenges you have ahead of you.