Oh God. Another noodle CX can make it really hard to be a customer

Two mistakes by the agent you spoke to:
1) You don't have to cancel all four tickets. Just the ticket of the daughter.
1a) This is accomplished by separating daughter's ticket into new PNR (something easily done in Amadeus), then cancelling that booking
2) Each ticket is US$120 not each booking. So "be charged $120 fee and re book" is incorrect (if you transcribed the call properly)

Case in point - just last month and also on BA - the Missus and I were originally booked HKG-LHR-AGP-LHR-HKG on BA as Cathay AM redemption last month. Two tickets in one booking.
After JAL announced rule changes to the JAL Global Club New status programme (launching in January 2024) , the Missus agreed to run for her own JGC membership by flying to Spain on JAL revenue. I will shadow her on my existing redemption (I am a pre-pandemic JGC member so no need to run. I took the luggage so she doesn't lose it during connections).
On Cathay AM side, I phoned to carve out the Missus from my booking, then immediately cancelled the newly carved out booking. US$120 (charged as HK$936).

Oh - and I changed my redemption to HKG-LHR-GIB//AGP-LHR-HKG. A bunch of AM agents said cannot do because HKG-LHR-GIB not available (suggesting married segment logic has to be enforced, at least for partners...), but I HUACAed 4 times until I found an agent who said yes.
Reissued (even though I paid a 38,000 AM repricing along with US$50) - I recovered the 38k AM somewhat by landing in GIB three times https://www.flyertalk.com/forum/brit...l#post35767789