Certainly a phone call to the Rick Steves Tour Department would be the best way to discuss your questions.
I very much doubt that they would accept a family of several people to participate in just a portion of a group tour, as the group dynamic is very much a part of the tour experience. From your perspective, you would be wasting a lot of money, paying for hotel nights, meals, transportation and other things that you would not be using.
However, to take advantage of the Rick Steves "team", I can think of two ways:
1. Use the "consulting services" at Rick Steves Tours to help you plan your own itinerary and experiences. There would be a moderate fee, but the staff members are excellent.
2. In Rick Steves guide books, including his Italy books, in chapters on major cities there are often recommended private guides listed, with contact information. Often these may be people who sometimes lead Rick Steves Tours, or serve as local guides for RS Tours in particular cities or regions. You might also get names and contact information from the Rick Steves Tour Department, maybe as part of a consultation, and could arrange some private tours for your family group with these excellent guides, assuming they are available for your dates. Good luck!