You could Google Hotel Primavera- which may have rooms in your price range. We stayed there a few years ago and it was about that then. The room was not fancy but clean , as was the bath. The breakfast was included in the price-and was awful- coffee ond rolls- so we would supplement across the street in a cafe.
Location can't be beat- right on the edge of Piazza Navonna- very nice and safe area..
Downsides: Hotel is actually two floors in an office building - there is another Hotel on another floor and some offices as well- this is somewhat common in Europe-we stayed in a similar situation in Germany. Also, they do not accept website bookings- you have to fax or call and you must send one nights deposit by check ( which was not cashed and returned to us on arrival.) Also must pay in cash- understandable with their low rates. There are ATMs and banks all around it, so not a problem. Do not accept tehir airport transportation- arrange your own- they charge to much. With those caveats, I think Vittorio might be able to help you. The location is a real plus!
There are a few reviews here on TA- including mine! Might work for you! If you can't find their fax, I'll look for it if you need..
Good luck! We loved our budget trip - stayed 8 days with Vittorio and had no complaints except breakfast and a coffee and cornetta are available all around the area, so no big problem.