IMO Hamilton is not the best area to be based out of, but that's just preference. Anyway, have you considered a vacation rental? They are a very popular form of accommodation in Bermuda. You can definitely find one for under $300.00 in June. A hotel (In Hamilton) may be harder.

Try www.bermudarentals.com for some affordable options. Another idea would be Fourways Inn. It is not in Hamilton but on the pink ferry route which would give you ready access to public transportation.
Have fun!

Also look at the Rosemont (not Rosedon) - right up the hill from the Fairmont, we have booked my (particular) in-laws there several times - rooms are nothing special, but have kitchenettes, it's primarily a business-traveller's home-from-home, nice pool, gorgeous views of the sound, gas grills on site for guests to use. Staff helpful about arranging transport and giving directions.

Bermudarentals looks great. Has anyone stayed at a unit? Not sure which ones are good.

Check tripadvisor for some reviews of specific properties. There are several regular posters who have rented from them and they have posted very good assessments. I have rented from the company and my experience was excellent.

We rented from them last June and had a very good experience. We chose a unit that was waterfront and was able to swim via the steps down into the water. The owner was very nice, it was a very affordable way to visit an expensive island. The owner was very kind and gave us things like a bottle of wine, some homemade jam, couple of bus tickets to get us started and some advice, speaking to a local resident is always very helpful. Another great thing for us was on the check out date, the owner very kindly let us stay until early evening, as our flight was at night. It really made a difference to our last day.

Odin, where did you stay?

We stayed in unit C23A last summer. It was a very nice property,a short drive to the south coast beaches and 10 minutes to Hamilton but it's located on a hill so it's more ideal if you are renting a scooter. It has reviews on tripadvisor under " Sunscape Bermuda" if you are interested. We have also stayed in St. George's various times. I highly recommend Samaritan's cottage ( it also has reviews on tripadvisor ).

GF74: I can't find the apartment anymore, I can only find the studio which was a separate building at the back of the main house. The studio is W26. There is a photo of the main house, and we had the apartment to the left of the photo, upstairs were the bedrooms and bathroom and downstairs was the kitchen and another bathroom, with access to the outside. Maybe its not for rent this year. I would not have necessarily recommended this unit, it needed a bit of refurb which Bermuda Rentals told us, but we loved the location on the waterfront. Also we booked about 3 days before we arrived in BDA, so that limited the choice considerably.
We paid a deposit via paypal and paid the balance of the rent to the owner when we arrived.
We prefer locations on the south coast or west, maybe St Georges as well and would definitely rent a unit again.