I agree with Canary that the Third Party pulled out whilst being overtaken by your daughter,and the TP actually pulled out when your car was probably at least two thirds past the TP.
This looks like the fault of the TP if the damages were to the TP front bumper and to the rear of your car.
If you explain this to your Insurer I am sure they will agree.
If they do not agree then it is up to your Insurer to negotiate with the TP, and I would add that any loss of use claim for hiring a car,or indeed paying for taxis, is their responsibility ,not yours.
This applies even if you only have TP cover and not Comprehensive,so I don't think that the person you spoke to knows what they are talking about here.
There could be an Excess on your Policy which MAY apply to the TP section,which you will have to ascertain by reading your Policy.
Hope this is of some assistance to you.