Yes where I work you can use your private health in the public system which often includes a private room when available. Unfortunately because its a public system your still not guaranteed a private room as other people may take priority (such as other private ladies who were there first or infectious diseases etc) You can not have your partner stay unfortunately (unless is exceptional circumstances) Your partner can stay in delivery suite but not the postnatal ward.

I would think early discharge would be the best solution for you, your DH and babe especially if you have good support at home and to help the reduction of illnesses and people touching your baby. I have this issue also and plan on discharging home within 4hrs of birth if possible. I'll be birthing in a public hospital as a private patient (as our private hospital only does c/s) and provided everything is normal I'll be out of there for the same reasons you mentioned (wanting privacy, not wanting the midwives/Drs touching bub and then other visitors coughing and ****tering in communal places)

I'll add you to my prayers that your next BFP is just around the corner and you have a brilliant birth and take home bub!