The key is to practice, practice, and practice again, so that going through the booking process is like a well-oiled machine. My friend and I have both managed it by using household accounts and having our credit cards on the system, and from searching for the flight to finalising the booking is less than a minute. (It feels much, much longer than it really is!) The process then boils down to search, click the flight, click the dropdown to select the other passenger, enter the CVV code and click next a few times. It's the sort of thing gamers would call a "speedrun"!

The other option of phoning will work if you can get through just before midnight and the agent is both willing to wait, and knows what they're doing - if you hit a queue and don't get through before midnight, you're stuffed.

There's nothing stopping you doing both, of course, both being on the phone (in case you get the timing exactly right) and trying to book online (if you're still in a queue on the phone)!