I'm sure others will have better suggestions. But my first thought would be to reward the good behavior. When he stops barking, give him a treat. Every single time he stops barking, treat him. Hopefully he is treat motivated. If not, give him what he loves (toy or whatever). Oh, and obviously you want to use teeny tiny treats so he doesn't get stuffed!
And your mom can definitely help you with this! In fact, anyone in the house can help. Consistency is key!

An idea for how he constantly wants to play with Hank is for you to tire him out so Hank doesn't have to deal with his puppy antics. If that is even possible to get him tired.

Good luck!