Barcelona sell out for this event and prices escalate. You need to take what you can find within your budget. I do not think you will find anywhere without heating.
Two years ago I spent my birthday in Barcelona - it coincides with my birthday AND with half term holidays throughout Europe. I booked my place 9 months in advance.
For more info consult booking engines and TripAdvisor.
For airbnb - use the map search function and the filters available.