Have you tried
www.vrbo.com? You select South Carolina and then individual condo owners advertise their homes and some are pet friendly.
We have stayed in Myrtlewood Villas (Myrtlewood Golf Course) is there and the public beach is about a mile away (they are between 38th and 48th avenue).
The Myrtle Beach website says dogs are allowed on the beach in the morning until 10:00 a.m., but when we were there the signs posted where you park said 9:00 a.m. for the summer.
Good luck with that website.