Budgies regardless if they are hand raised or not need to be on a safe and stable environment. Since you have multiple pets and small children, your budgies need to be on a safe room where the dogs/cats aren't allowed in.
When it comes to young children, it's our job to supervise them closely and to educate them on how to communicate and interact with a given pet, in this case a pet bird.
If there is so much chaos that both of your budgies show signs of stress and this is something that you are unable to change, then thinking on the budgies long term happiness and welfare, it would be best for them both to be rehomed.
For your current situation with consistent work (this means daily interactions and training sessions), lots of patience and understanding, it's possible to finger tame the shyest and most nervous of pet birds.
I don't think it would be fair for both of your budgies (since they are bonded and good friends) for you to rehome Rubble and to give Skye a potential new friend which may or may not be a good fit.
Even for hand fed and tamed young budgies, if the proper work is not put into them, the taming achieved can also regress and you will end up with the same problem.
Since your Skye was making good progress with you in the past and he showed more promise, you can start to focus more on him on building the trust he has in you and to have one-on-one training sessions with him (on a safe room where no dogs or cats are allowed in).
Later on, you can use the bond you have with Skye to make Rubble more receptive and open to you. I have used this method multiple times with success where I have used a fully tamed budgie to earn the trust and tame another budgie.
For all the detailed information, be sure to check the stickies at the top of the Taming and Bonding section's main page.