Don't give up hope this is usually a long drawn out process but ultimately very rewarding in the end.
I have gone the St Louis route and they were able to give me my Uncles unit information but that was it.
ONCE you have this you can keep following up with alumni associations and veterans etc.Keep on top of the St Louis record office as I have determined that there were many units in New Guinea (Probably to many to narrow down who your dad served with)
I will send this off line.In the mean time you need to exhaust all of your family resources.
How about the VA??Did you dad apply for benefits or need VA assistance at any time upon his return?call the VA (phone: (800) 827-1000) to inquire on how to obtain this information.