You might consider doing this the other way around?
Book the 3-hour tour of museums and basilica offered on the Vatican's website. It hits the highlights of the museum, and enters the basilica from a corridor at the back of the Sistine that's reserved for tours, thus saving a long walk around to the church and standing in its security queue. When the tour is finished, you can freely wander St Peter's for as long as you wish.
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni/musei-vaticani-e-basilica-di-san-pietro/visita-guidata-musei-e-basilica-s--pietro-per-singoli-e-gruppi-.html
Tours start at 8:30, and will be available for the 21st of April on the 21st of this month
Another popular option with RS posters is the Walks of Italy "Pristine Sistine" tour. It's more expensive BUT it starts in the Sistine before the general public is allowed in (7:30 AM) thus giving you the opportunity to see the chapel without its usual shoulder-to-shoulder mob. This one also accesses the basilica through the back corridor.
https://www.walksofitaly.com/vatican-tours/pristine-sistine-chapel-tour?gclid=COC3mIa4hdICFcm3wAodGhEL4w
Both options eliminate standing in long ticket queues at the museum. Otherwise, just book a timed-entry slot to the museums, although 9:30 might be advised in case the security queue at the church is long (get there a little early). It's very large and there's a lot to explore.
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni/musei-vaticani-e-cappella-sistina/visita-libera-musei-vaticani-e-cappella-sistina.html
If you take one of the tours, have your hotel hold your luggage (or any other items not allowed inside the church/museums) as you won't be returning to the museums from the basilica.