Jenny, I would DEFINITELY book an early entrance tour that includes additional museum treasures + basilica. It's going to be hot in the museums in June, and the crowds will be IMMENSE. The best of these tours last 3-4 hours or so, get you into the Sistine just before it's hopelessly mobbed, cover the highlights in a structure than can be difficult to navigate on one's own, and get you into the basilica through a back entrance reserved for tours. That right there saves you a long-ish outdoor walk between museum and basilica + a potentially long wait - in the elements - in the security queue for the church so it's an efficient way to do both. You can linger in the church as long as you wish to when your tour ends.
While there are parts of the Vatican Museums that are less visited than others, so the hordes can be escaped for a bit, with pre-teens + summer heat + summer mobs, I think a museum+ St. Peter's tour is the best choice. I wouldn't do either of the tours listed above as #2 looks only to get you in the door early and doesn't cover the basilica, and #1 doesn't cover any highlights in the museums except the Sistine: IMHO both are sort of rip-offs for the price.
While usually a little more expensive, do take a look at the Walks of Italy "Pristine Sistine" tour Christine mentioned? This one has been a very popular and very highly-rated option amongst RS travelers to Rome. It is also currently on sale so you'd land a high-quality tour for a bit less than the most expensive City Wonders option you were considering.
https://www.walksofitaly.com/vatican-tours/pristine-sistine-chapel-tour/?gclid=EAIaIQobChMIoJL4hKaF5gIVih-tBh3FawvqEAAYASAAEgI6evD_BwE
One more note? Ignore any mention of museum visitors whom are not on an organized tour sneaking into the basilica via that back passage. While guards at that door have been somewhat lax in the past, the word recently is that they've gotten much more strict about checking badges.