Traveling with a larger group (and with children)

If it were me, I'd start by looking at large apartments to rent via airbnb or vrbo or some other rental site. You have a lot of people and staying in a hotel will be very costly...that's at least 3 or 4 rooms.
I did a quick search on airbnb and found a beautiful apartment right near the Pantheon that sleeps 8 with 8 beds for $367 per night this June. I'm not sure when you're traveling but for two families it would be perfect! It has lots of positive reviews. That's all I know about it but there are quite a few others in the area that look like they would fit the bill!

I second the vote for the apartment. It gives you a private gathering point for mornings and evenings. You can all trickle out into the living room area at your leisure, have coffee, plan your day, wait for the kids to be ready. And then, at night when the small kids are asleep, you have room to relax as well.

It's not that hard to find an apartment to sleep 7. I use vrbo, generally, and you can search by the number of people each unit will sleep. I go for units that have lots of reviews (at least 15, hopefully) AND the reviews are recent and specific. I'm not interested in the "great place" reviews...I only count the detailed ones. Next, I send out a few queries. I nix anyone who doesn't get back to me in about 24 hours, which sounds harsh, but I'm pretty careful. I'm about to do a business transaction with someone thousands of miles away and I'm going to have to pay them a deposit, which can often be quite hefty, so I want work with the most organized and efficient of the owners.

You can check booking.com for apartment listings; you will not have to pay a huge deposit in advance.