Sometimes I try www.rome2rio.com to get my mind further out of the box. In your case it showed some other cities in Italy with an added train ride, plus that other Rome airport (Ciampino).

Kayak sometimes turns up some odd options, when it hacks together legs from different carriers.

When you are searching, if the particular venue has the "nearby" option, it can be worth clicking.

http://matrix.itasoftware.com is very powerful, searching on a month of dates and letting one look at nearby options (varying the range). Also will check specific dates for +- a day or two.

For smaller odd points, sometimes I just do a search on "fly abc - xyz", and that can pick up different airlines.

Hotwire has their "Hot Rate" deals, if you can cast your fate to the winds as to specific departure times and airlines. Too bad Priceline doesn't have the flight bidding any more, as occasionally that worked.

There is the backwards approach of looking for cheaper flights from Rome, and finding one reasonably close to MCI.

I'm sure others here will have some more ideas, but the above might help a bit meanwhile. Your logic and approach is already good - open mind and willing to explore. Getting to competitive points and using deals from there.

I don't know anything about group rate tickets, but maybe somebody else does? I'm thinking 8-10 might not be a large enough group?