Check out Samson Realty. They manage rentals for a lot of home owners in Providence: https://www.samsonrealty.com/

If they can't find you one, call up a big rental group like Bilodeau or 02908. But Samson was able to find me a Wayland Square studio in my budget and cheaper than the rest of the city a year ago and they will be honest and point you elsewhere if they can't find it.

However, there are a lot of 2BRs in Fed Hill/West End for 50-60% of your budget.