CX and QF operate from the same terminal at BOM (Terminal II-A) and there is a transfer desk inside the sterile area. I don't believe the Indian Government requires you to have a visa as long as you don't mind spending 6 hours in a stark empty windowless corridor. You will not be able to access the lounge, since it is a third-party lounge and access is by invitation only and invites are only issued at the check-in counter which you cannot get to.

However, the specific airline policy may be different on this issue. I know that AI for example requires all pax to possess documentation allowing entry into India, even if they are only going to be in international transit. However, since AI and IC are the only carriers that have online international transfers at BOM, this may be an airline specific policy.

You should check with QF in advance to verify that they will allow you to board at SIN without an Indian visa. Since it is a oneworld connection and they will probably issue both boarding passes at check-in, you may get away with it.