Unfortunately not all births were registered and as both Debra and I have commented, Katie's marriage cert may provide more information.
Yes interesting there is no mention of Katie in Agnes's death notice previously linked by Sue, but I bet the 'brothers' were more than willing to accept the bequest from Katie's estate

Of course information in death notices is not always correct just as information on death certs is not always accurate eg Agnes's age should have been 45 years not 40. I have a number of death notices of my ancestors and not all the family were mentioned.
I don't know how anyone can research a family without purchasing a few certificates along the way....just my two bob's worth

I assume you are aware you can purchase images of Victorian certs online and immediately download for AUD$20. You mentioned you had found an illegitimate birth previously....perhaps when you locate it in your papers you could post the info please.
Have you searched in NZ for Agnes and Archibald's arrival? I was fortunate to find some of my OH's rellies arriving in NZ but no passenger records leaving Aus.
Hopefully some more info will be found about Philip H.
Cheers
Cando