For example, on realrome.com one of the larger apartments in Testaccio is showing availability for your dates. A bit outside of the center, but it looks like a nice place, big enough for your family and WELL within your budget. The Reginella apartment also shows availability on your dates, great location, 130E/night for 5 nights 4 people.

Of course it may be that they haven't updated their online calendars, but you never know. And if you are able to spend less in Rome, you could spend more, say, in Venice.