Features vary a great deal from one model to another. It’s always a good practice to read the user guide for a device before buying it and make sure you understand what features it offers.
The model you have is a basic siren/strobe. It does not offer a beep or any other alternative sounds. So I don’t think it’s going to meet this particular use case. ( it would freak out both your pet sitter and your pets, essentially the same as a smoke alarm going off.)
https://support.smartthings.com/hc/en-us/article_attachments/115000454183/UT_IndoorSiren_manual.pdf
Do you have an echo device of any kind, including echo.?
Amazon just added a feature last week where you could now have the echo play either a sound file or speak a notification, and that can be integrated with SmartThings. So you could have the echo say “the system is now armed.“ or just play beep.
Alternatively, a lot of people use a small indicator light to indicate whether the alarm system is armed or not. There are a number of options for that. See the following thread from this week:
Otherwise, since I believe you just purchased this siren, you might want to return it and get one of the ones that can play multiple different sounds.