Could be very difficult finding better-class hotels at that time of season for that price -- especially if you want to have everyone in one large room. A hotel like Porto Mykonos would cost you over €700 per night if you want something more luxurious and upscale.
Also, you probably will have to contact hotels directly to inquire about availability for four adults, since many hotel booking sites don't provide for 4 persons unless two rooms are booked, and some indicate large suites are available but don't specify how many adults they can accommodate. Rochari hotel has doubles for €148 per night and a suite at €220, for instance, but the website doesn't say how many people the suite can sleep. Many hotels, like Ilio Maris, show triple room availability, but don't indicate if you can add a cot or extra bed for a fourth person.
Here's a link to a local Mykonos accommodation website that has listings for many (but not all) properties on the island. You can compare Town to beach locations as well as compare hotels to studios. You might find something in your budget that doesn't have a pool but will include breakfast and wifi.