Go with the Argentine passport Rob, and get your application in a.s.a.p, like today!
Unless you have already applied for Australian citizenship by descent for your baby, 4-5 weeks is not enough time to process the application. If you have any queries contact the Australian Embassy in B.A. Dual nationality is the way to go these days, plus it will give your child a world of choice in the future. Just be aware that if there are any existing prerequisites in the country of permanent domicile (military servce etc) they will take presedence over the alternate citizenship. Was a migration agent in a past life so I know my stuff.
Have a great holiday.