Generally speaking, budgies aren't that fond of being petted and by trying to pet them against their will, this will negatively impact the trust they have in us and the bond will suffer a setback. The biting is BB's way of telling you to back off and that he is being bothered by the petting.
Taming is all about earning the pet bird's trust in you and for this you will need to learn how to approach your budgies, make a connection with them by using your voice and to respect the safe boundaries set by them.
Be sure to check the information on the stickies located at the top of this section of Talk Budgies, so that you will know how to work on forming a long lasting bond with your budgies.
BB is not a bad influence, given the fact your new budgie is in quarantine, it's only natural for them to call to each other and to also screech on occasion. They know of each other's presence and are excited to meet in the future, once quarantine is over.
You can work on individual training sessions for each of your budgies so that they are only focused on you instead of being distracted with each other.
Good luck with everything and embrace the singing, the occasional screeching and flock calling.