The cruise company appears to be offering tours to the Vatican,
Coliseum, etc but the local private companies tell us we 'won't be
able to get near them' or they will be closed.
You've a couple of issues here with Sunday, June 2. First off, the Vatican Museums are closed all Sundays but for part of the day on the 4th Sunday of a month. The National Museums - such as the Colosseum - are free on first Sundays of a month: definitely NOT a day to try and do Rome's most-visited attraction. To my knowledge, tours are not available at some national museums on First Sundays: this is true for sure of the Colosseum, anyway.
There has been talk recently of the Italian government doing away with Free Sundays but I've not seen any news on that lately.
But there are other attractions which will be open. For instance, if you jump on reservations as SOON as they're available for June 2 (abt. 3 months in advance; they will go VERY QUICKLY!) Galleria Borghese will be open plus still observing their excellent crowd control, even tho it's a national museum and free (aside from small reservation fee) on that Sunday.
Ostia Antica is another possibility although it's not "in" Rome. Does your ship offer a tour to that one? Although it's a National Museum and free on First Sundays, it's unlikely to be as heavily visited as some attractions in the historic center of the city. If you can find other shipmates interested in this one, it's possible you could split the cost of a private driver.
Or via train, although it's a little messy:
A number of churches in Rome should also be open for visiting between Sunday masses, and many guidebooks provide self-guided walks you could do on your own, taking in the architecture from the exterior of those landmarks either closed or overrun. Just a start?