Room or Studio Wanted

We are a 25 year old professional and a 1 year and a half smart male explorer. I work in a bank and I need a room for myself and my son for one month (starting end of July/beginning of August), the time to find a house in London. We would like to spend a maximum of ?425/455 for the room for one month. I would prefer no deposit since it's just for a month. My son doesn't cry at night and sleeps all night long. We will be out of the room for most of the day for work and going to the park/journeys. I would prefer Croydon and South London area (Tooting, Morden, Wimbledon) or south east( Lewisham, Crystal Palace, Beckenham, Greenwich) but I'm open to consider all areas of London as long as it meets my requirements. Please contact me if you are renting a room that suits my requirements. Thank you and looking forward to hear from you. Kind regards, Flavia