For moms who've had babies in the summer (heat question)?

I am due on August the 12th and I live in LA. It gets REALLY hot around this time of year starting in July and ending maybe in early October. I am concerned because my bedroom and my babies bedroom is upstairs. We have an old house and priced an air condition system but it was just TOO expensive, the way our home is built we would have to have 2 seperate systems and pay double (one upstairs and one downstairs). My question is what other alternatives do I have? Right now, we are looking into getting a window air conditioner for my room (the baby will be in a bassinett) and one for his room as well. I’ve purchases an HEPA air purifier for the baby’s room and my room so far, but what advice can one give me keeping my baby cool at night while he will be sleeping (heat rises in our room). We can open all our windows, etc. But I don’t want my baby to get over heated during these summer months.