Since Broly had turned good, here is how an idea on what to do with him:
Similar to "What if Broly Turned Good" by Masako X, I suggest that Vegeta and Broly to go to the Gravity Chamber and have Broly train with Vegeta and train together so that Broly would master his Legendary Super Saiyan form.
This picture shows that after Broly had nowhere else to go after he prevents the Majin Buu saga, Vegeta decides to take him in and have him live in Capsule Corps. After that, Broly becomes Vegeta's servant and a sparring partner. The reason why Vegeta takes Broly in was because Vegeta is the prince while Broly is a subject. If he does that, then Broly would use Legendary Super Saiyan power against Beerus:
This picture shows that thanks to Vegeta taking Broly in and training with him, he can control his Legendary Super Saiyan power and he hits Beerus. Because of this, Beerus had to use 20% of his power to take Broly down. Also, if Vegeta had any plans on bringing Broly to Earth or Broly ask Vegeta to bring him to Earth, Vegeta will have Broly take a tour on Earth and would teach him about the Earth's culture and have him train in Martial Arts. Also, if Vegeta did this, then Broly's heart will become pure and pure enough to join the Super Saiyan God ritual. Also, if Vegeta and Broly's relationship is well developed, then there relationship will become the male counterparts of Kale and Caulifa where Vegeta is the Caulifa to Broly's Kale as well as having Broly's personality closer to Broly from Super. Lastly, if Broly is well-trained enough, he would have mastered his Legendary Super Saiyan form and will achieved his Legendary Super Saiyan 2 and has the possibility to become a Legendary Super Saiyan Blue:
This picture shows that if Bringer of Death gets to Super and Broly survives the events of the Majin Buu Saga, this is how there relationship has evolved. Also, this pictures show both Broly and Vegeta working together to defeat Golden Frieza.