Yes it is doable, not easy but doable.
The tricky part will be the meals. My sons had baby food but only home made and when I tried to give them jars (no pouch at that time) they always refused.
I think you would find it easier in a luxury Italian or Spanish holiday than Australia or NZ. I live in Sydney and I don't think it would be easy for you to eat out and have the restaurant prepare you something for the baby as well, whereas in Italy or Spain any real restaurant not the crappy one with pictures of food, would be able to prepare a risotto, chickpea soup , pasta in broth, a streamed fish and so on.
There are wonderful luxury hotels in Sicily where you would be treated like a queen and both Italy and Spain love babies and would go the extra mile for them. Here in Sydney, it would be I think more difficult to accomodate the baby's need.
This is really nice www.hotelvilladiodoro.com/eng/restaurant-diodoro-taormina.html and taormina is magic.
We have also been to the Bahia del Duque thetaishotels.com/bahia-del-duque/ or www.iberostar.com/hoteles/tenerife/iberostar-anthelia
We went to the Meridien on Iles des Pins in New Caledonia which is so luxurious and exclusive but no Medical service on the island so you shouldn't go anywhere without a fully serviced hospital, and not only a medical point mostly serviced by military doctors
Outside the meals issues you might need to consider how knackered you will be after the long haul flight even in business, because you might sleep but there is no guarantee the baby will and it means starting your holiday exhausted.
or, go for the typical luxury resort. We went to the Barcelo Bavaro in Dominican Republic when DS1 was 9 months old BUT he got sick. There were 10 restaurants including a 24 hours one and we managed to find something for him to eat but it wasn't easy and then we had ants in his milk powder. so nice but tricky. I think the total now has changed but Barcelo is a good chain to look at.
Consider your loneliness as well. In Italy and Spain, people talk more to strangers , our previous postings were in Italy and Spain so I know.
Australia is beautiful, people are nice but less warm to strangers, whereas in a tiny square in Italy, you exchange a smile and words.
Too bad, Rome is so chaotic and Venice has so many stairs.