Rome B&B or apt for a family of 4

Try searching on booking.com.

I can give you two recommendations from personal experience. The first is a lovely large two bedroom apartment in the area of Trastevere, not too far from the Vatican. We were two couples that stayed there in 2013. It was perfect. I did a lot of website searching before we decided to go with this apartment. Two good sized bedrooms and bathrooms, good living room and kitchen. Wifi. AC. The owner was very helpful and replied quickly. She and her daughter both speak English. She also arranged for airport pickup for us. The apartment is very well located. From there we walked all over and it was about a ten to fifteen minute walk to Vatican, not very far. Lots of restaurants nearby. The apartment is on the ground floor on a beautiful old street, so no steps to deal with. Look up Residence La Scala in Trastevere. Trastevere is a very charming neighborhood. The apartment has its own website. I have recommended it to friends and I would stay there again. I checked its website and the prices are very reasonable for what you get and are right at the top of your budget.
Several years ago I also had a room for five at Hotel Des Artistes near the train station. I took three of my kids and a neice and they had a room for all five of us with a bathroom right across the hall. I believe they now have both a hotel and a hostal section. Nothing fancy but adequate and budget friendly. Within walking distance of Termini train station. You would have to check out its current reviews but it worked just fine for us but that was several years ago. It was one large room with a double bed and three or four twin beds. I don't remember if it had AC. The apartment option is much nicer and is in a better area.

http://www.casavisconti.it/Index_EN.htm

Near the Vatican, my friends and sisters stayed there under my recommendation and loved it!

Our first trip to Rome (family of 4) we stayed in our first European Apartment and LOVED it! They have so much more room than a hotel, and everybody could spread out and have their own space. 4 people in 1 room (even if it's a quad) can get really cramped. We rented our apartment from this wonderfully friendly family. At the time they rented just 1 apartment and they lived next door in the other apartment. Now they rent both apartments as they've moved to a different address.

http://www.vrbo.com/305032

It was a fabulous decision for us even though we were very wary of rent an apartment.

It is late for booking July. Try VRBO.com or Booking.com. The latter will also show B&Bs and hotels. Don't fret too much about location, but Prati (near the Vatican) has a lot of options in normal circumstances and is more convenient than many people realize.

You might also check Cross-pollinate.com. They seem to have several available in your locations.

As for Cross-Pollinate the only problem I have with their properties is that there is no AC so summer can get oppressively hot trying to sleep in unairconditioned rooms with the windows closed (to keep out the noise and the insects). Rome never sleeps; if you try to sleep with your windows open YOU will never sleep right along with Rome!

Just to clarify...you have the option to choose AC as one of your requirements.
w/cross-pollinate

Sorry Tim didn't make that choice.

www.homeaway.com is also a good site for apartments in Italy

Thank you everyone for the thoughtful replies. We found a great place to stay!