Hi, don't know if this helps, and I caveat I do not know how early your flight is - but try the Hub by Premier on Rose Street? It is in the New Town area - fabulous location! There is a trolly from around the corner we took from hotel directly to the airport the day we left. 30 min tops door to door, rolled on and rolled off with big bags. Short walk right into terminal. Could not have been easier. I do not know where Hyatt is, but sometimes those airport hotels require a lot of walking to terminals. Plus, don't you want to maximize your time in Edinburgh by having a last (early) night in town vs. the airport?
HUB room was $134 USD per night with breakfast. With a one year old, get the "bigger" room (they have small and larger rooms). Located in a pedway so perfect for a stroller.
Here is link to HUB:
Here is link to trams - you will need to check what time they start running in the morning and whether that will allow you to make flight.