Michael,
St Kilda isn't that far from the sandbelt - The National and St Andrews Beach are not the sandbelt and are an hour and 10 minutes away - and you are going against the traffic so don't worry about that.
If you can find apartments in Cheltenham, (Victoria, RM)Sandringham, Brighton, Mordialloc (Woodlands), Oakleigh (Metro,Commonwealth Yarra) Hampton or Black Rock (RM) try those suburbs but St Kilda would be more fun and you are much more likely to find apartments.
You could stay in the clubhouse at Peninsula G andC.C at Frankston. It is a terrific place to stay and much closer to the Mornington Peninsula. Probably it is 25 minutes from RM and 40 from The National.
I live 5 minutes from St Kilda so you may be able to scrounge a few rides with me if I am around.