We are moving out of a fab doggie friendly place in Fullarton (Gumtree ID 1006106287). Dont let that put you off as not near the beach, bus stops right at the bottom of the rd and takes you straight to Glenelg. You are 5 mins from the city, right by the hills and there is a dedicated doggie park at the top of the rd.
Also the owners would be open to offers on the weekly rental. Bit dated but huge and fab area.
In my search for a new house I have found that anything that is new and nice is not doggie friendly at all. Or if you are prepared to compromise on area, you can get one. Try looking further south if you want pet friendly and modern.
Good luck xxx