I stayed in the historic North End. When I booked I saw TD Garden was close enough, but didn't actually realise until I got there just how close it was. Probably took us 5-10 minutes at most to walk there. We stayed in an AirBNB and from memory it would have been around 150-160 dollars. I don't think you're likely to find anything of decent size for that price. But if you can handle a bit less room it really isn't that bad. Particularly if you're going in March, it will probably heat up a bit by then so you don't have to stay in your place as much. Where I stayed was right near a part of the Freedom Trail too.
The North End also had a lot of good Italian restaurants, a few bars and was a close enough walk to some seafood restaurants including the well known Union Oyster House.
Can PM you the place if you want.
Mate that’d be fantastic if you could