Find 2 CW or F redemption seats for youselves to NRT/HND/ICN so you can use your 2-4-1.
Buy a full Y seat for your daughter which you can Uua to WT+,then try a day of departure buy up to CW.
This assumes,1 The company will reimbure your daugters Y seat,
2 You are able to arrive up to 2 weeks before commencement day for your daughters new job.

If you are short of a few avios,you can always buy a few from BA.Not great value,but useful if you need a top up in a situation like this.

Look at taking out a credit card that will give you a sign up bonus for the extra avios.