I have a cat and a small, 24-foot RV. Three years ago I was here in Mesa in August. The outside temp got up to 111, and inside got up to 120 (no cat at the time, by the way). I turned on the air conditioner, not realizing that the fan had a high and low setting. I was VERY new to this. Anyway, I could not get the temp in here below 115 with the air conditioner running for hours and the fan on low. I was doing some things wrong and was in direct sunlight. It was cold by the ceiling and hot half way down. Imagine the heat down at pet level. In moderately warm temperatures where it'll get up to 90 or 100 in here, I leave the air on in the cool morning. It'll kick itself on when the temp rises. Sometimes I set the air conditioner fan to run fulltime to keep the cool air circulating when the compressor is off. In cooler weather (between 75 and 80) I let the air conditioner run the fan on auto so the fan stops when the compressor kicks off. However, I haven't had Missy in here in Arizona heat. I don't think I'd leave her in this little place, at least not without some kind of shade and other preparations. She's a cat, so she's asocial and afraid of everything. Have you considered doggy daycare for your dog? It's only for 8 months.
BTW, I have tried keeping both Fantastic Vents open, one pumping in air, both windows open, and the back door open with screen closed. It can still run up to 90, on a warm (not hot) day. Leaving windows and doors open invites breakins in some places and certainly doesn't keep out rain.