Taking client calls after hours using my personal phone

Veterinary corporation I work for wants us to take turns being “on-call” after closing hours M-T only for current clients. The phone calls get automatically diverted to an answering service, which screens for people that are actually clients or just randos trying to avoid taking their pet to an emergency clinic. For an actual client the answering service texts me with minimal information : client name and phone number, species of pet, and extremely brief summary of the problem. My issue is that I have to remember to turn off my caller ID before I call them back. When I don’t, said client has my actual personal cell number which becomes a PROBLEM because so many people don’t understand personal/professional boundaries. To my knowledge there is nothing explicitly written in my employment contract about this, but I feel like I need negotiating points on this. Any thoughts?