Here are some specs on the hounditioner.
There are some immediate problems. It requires 120 volt AC at 350 watts. Its EER is 4, even without considering inverter losses when operating on battery power.
The Zero Breeze is the small type of A/C your pet cooler idea needs. If its reliability proves itself, it would be a dependable and efficient way to implement your idea, as it only requires 12 volts DC at about 140 watts. Not a big battery drain over a few hours.