Yes! We included both of our dogs (6 years and 6 months at the time) and had my MIL watch them while we were taking pictures without them - they love her and she's a dog whisperer so it made the most sense lol. We didn't get anything special for them to wear but we're really low-key in general. One piece of advice is get there early and let your dogs explore - ours were so psyched to be out and about that they were hard to wrangle for pictures!
We included our dog Lilly in some of our photos. My mother-in-law tagged along to watch her when we were doing the photos without her, and it was super helpful! We just chose a simple collar and leash that would coordinate with our outfit choices without drawing attention.
Haha, not engagement photos, but my BIL included the family dog in their getting ready photos on their wedding day! They are the cutest, with little mate sitting at the front posing perfectly. I think bringing along a friend to pop in and take doggo when you've got the shots you like would be a good idea, and if I had a pet, I'd definitely want at least one or two photos with them!