Whichever the process (online or ticket office) it's UL that you need to deal with as it's their program. They will issue a ticket on CX subject to availability on the dates you choose, but CX otherwise has nothing to do with any of it.

You could use the BA online tool to check for availability before contacting UL so you already know which dates are available. Outside of holidays CX C class is pretty easy to get within Asia.