Upvote:2
Since you already submitted the paperwork for the baby's ETA application, you shouldn't put in a new application.
Australian visa applications can't be expedited, and the embassy can't assist with visa inquiries.
It's likely that the baby's ETA will be approved in time for travel. However, I would encourage you to look at your airline's ticket change/refund policies in case your baby's ETA isn't approved.