I've been booking infant tickets for the last couple of years, so I have some experience with this

It's normal that you can't do infant bookings online ex-HKG. Needs to be bought on the telephone with CX reservations or through a travel agent. I don't know why.

For travel starting elsewhere, you can buy infant tickets online.

My suggestion - don't buy the infant ticket using Asiamiles. You can book both adults on Asiamiles and only buy the infant ticket using cash. That's what I'd do. My travel agent can issue a standalone infant ticket, and link it to a ticket that's already purchased elsewhere, so I suppose that CX reservations would be able to do the same.

Actually, if you're not bothered about bassinet seating, you don't even need to buy a ticket for your infant in advance. You can even do it at the airport in HKG, but that's probably more hassle.