Suggest to look into the systems that use the RFID type collars or the dog's microchip.
When the system senses your dog's chip, the door will open. This type of system will allow only select pets to open the door. You could allow the dog and prevent the cat.
RFID systems are battle tested and very tough. Any product you buy in a store is typically equipped with an anti-theft tag. These are RFID tags. Microchips in pets use the RFID technology. The RFID chip is a passive device. It only returns a signal when excited by the systems range and frequency.
If you are really hard pressed to find a proper door, then there are options to make an RFID controlled door using the Adriuno software coupled with a couple solenoids to control the door locking.
Some people are installing the Adriuno system on motorcycle ignition system. The rider scans a fob to "unlock" the bike. Motorcycles live in a pretty tough environment, wet, heat, cold, vibration.