Hi and welcome aboard!
Whilst I sympathise with your requirements, the problem here is not the price as much as the short rental. Most villa owners will only consider bookings of 3 days or more simply because of the housekeeping costs of cleaning and turning-round the house after 1-2 days stays. Worse still, other than at very short notice, most villa owners will not consider such bookings because it blocks the possibility of longer bookings. Imagine how they would feel if they agreed to let you have the place for a 2-day weekend and then were offered a 14-night booking?
So, I'm afraid it is simply a matter of wading through our lists looking for properties in the right price bracket and with the right features, then contacting them hoping that they might consider your request.