Ah, OK - I read this as "is it OK to stay the night with your ex, or would your current partner be justified in being upset by this?"
When actually your question is "my daughter's father who has never met her before can finally be bothered to fly over and meet her. He wants to stay at mine - DOES THIS MEAN HE'S SINGLE??"
Basically your ex is too tight for a hotel and thinks it would be fine to just mooch off you for a bit. And rather than thinking" wtf, this guy has no boundaries or any sense of how this might affect his daughter" you're thinking "hurrah, this must mean he's single, maybe he'll see how brilliant living with us would be and suddenly become the partner and father I've been longing for".
You aren't wrong or stupid to want that OP - being a single parent is hard, you loved this guy and it must be really tempting to day dream about you guys getting back together. But please protect your heart and protect your daughter from this guy letting you both down again.
Don't sleep with him. Don't let him play house for two days then fuck off again. He may realise what he's missing out on, decide to relocate and become a family with you - but there's actually nothing you can do to make that happen. If he's going to do that, he'll do it whether he's sleeping on your sofa or staying in a bnb. The difference is that you'll feel a lot less hurt and destroyed when he does push off back to Australia if you haven't spent the last few days sleeping together, making each other cups of tea and generally pretending to be a couple.
You'll be looking at your daughter playing with this guy and thinking "there's no way he isn't falling totally in love with her - and look at how happy she is with him. He'll realise how great she is, and how much she needs him, and he'll stay forever". Whereas there's a strong chance that he's thinking "Wow, what an excellent father I am. Clearly there's no need for me to feel guilty about not actually being in my daughters life - our current plan of Skype chats and nothing much else will do nicely. when I go back to Australia I can spend years telling everyone how close I am to my kid and how much she means to me. Even getting a bit teary about how much I miss her. DD is fine with it, OP is fine with it - I'm basically father of the year and an excellent person. LATERS DAUGHTER - maybe I'll see you again in a few years".
Good luck, op 