The incident occurred close to her residence after she was coming from her morning swim on 14 July.
Another woman, Gisselle Javier, was also mauled by a pack of dogs on Rock Hole Road, George Town, on 24 July.
“It would really look bad on the Cayman community,” said Mr Jairam.
“Cayman has a lot of animal lovers and I’m sure they won’t agree with going ahead and shooting these animals.
The Blue Iguana Recovery Program has worked through challenging circumstances to bring back the population from near extinction and at the rate that these animals are being killed, there will be no breeding animals remaining in the park in a few weeks,” said Mrs Ebanks-Petrie.