The main partner you’ll take advantage of is Delta which you can redeem your Velocity points on. I’d suggest departing Barcelona or Rome as the other cities have one of the highest taxes and fees in the whole continent as outlined in this pointhacks article.
If departing from Barcelona, you could take advantage of Delta’s service to JFK or Atlanta and connect to Miami. From Rome, you have more options. You could leverage Alitalia as an often unkown partner which offers direct service from Rome to Miami or again use Delta via Atlanta or New York. Note the different aircraft types on these routes though. Delta usually uses their 767 on these routes which have poor business class products but if you encounter their a330 aircraft, it will have a much better reverse herringbone product. Economy wise, Delta aircrafts have newer cabins in general.
Miami to Sydney can be covered by Delta and Virgin Australia themselves. You will have to fly to Los Angeles on Delta and then connect on either airline’s services to Sydney. Given the lack of business awards on these routes, you may end up having to transit through Brisbane or Melbourne.
A family of 5 is tough to book in Business so manage your expectations. I suggest using Delta’s own website to search for availability and look for space with the ‘Lowest Price’ tag on it. From there, you can use Virgin’s own website to confirm. Often though, they can’t search many oddball city combinations. It’s good to call up to check and confirm the space is available and book over the phone. They should waive the booking fee as well.