Domestic check in should be fine. I don't know if they interline to EY, but that's a separate issue.
In terms of the airport cost. If you use Opal that gate fee is capped for the week, so you'd only pay $23 towards that, with no gate fee required when going back to the airport from the CBD. There also used to be a weekly gate pass. but I can't find the pricing. Might be worth contacting airport link.