You might also consider the 3 hour museum/Sistine/St Peter's tour offered on the Vatican's website. It's more economical than the "Walks of Italy" tour - which gets great reviews but would be expensive for 4 - and it gets you into the basilica directly from the Sistine through a passage reserved for tours.
The museums are vast so a guided tour will steer you to some of the highlights, and you wouldn't have to deal with another security check line to get into the church. When the tour is finished, you're free to stay and explore more of St Peter's on your own for as long as you wish to (but can't return to the museums.).
You'll find it here:
https://biglietteriamusei.vatican.va/musei/tickets/do?action=gototipovisitapertag&tag=museiebasilicavgsingoli&weblang=EN
It's too early to book for June yet; advance reservations will be possible 2 months before the date of your visit so the corresponding date in April.