I don't know their rates in May, but we have stayed in the Colonnade Hotel, next to Copley Square, which is lovely. If it's warm, they have a nice rooftop pool, free to guests. The rooms were a good size for the price.
This is college graduation time, so you will likely run into problems with availability and prices to match. You didn't say specifically when in May you are coming, but if you are flexible, try different dates - or even switch to June if you can't find hotels for your price point.
I`ve stayed in the Colonade and also the Hilton Back Bay. Both excellent.
Does the $300/day budget include expenses, or just lodging? If it's just lodging, I recommend the Copley Marriott. If you are looking for a more cost-effective option, I recommend the Holiday Inn Express across from the TD Garden. I know the state's tourism website massvacation.com/find/lodgin... has a ton of Massachusetts lodging options, definitely give it a look.
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