Daphne Inn looks beautiful and is listed in several guide books as budget, but their rooms are still high now for your budget ($300). I checked a few weeks ago for the beginning of October.

the Hotel Parlamento www.hotelparlamento.it often gets good reviews, and the location is good. I just checked their rates, their doubles at their lower end do fall within your range, though probably they are small. I also vote against Termini area. There are some acceptable streets in the area, but overall not an area that appeals to me.