AFAIK it’s only classed as being for hire or reward if you’re being paid for doing the driving. As you’re contributing to the cost of the trip by paying for yourself (and daughter) it’s not a commercial arrangement so IMO you’re effectively simply driving a bunch of mates to an event.
It’s the same as you driving a few people to an event in your own car where the passengers contribute to the cost of fuel, the only difference is that in this case you’re collectively hiring a minibus to travel in.
Hire and Reward doesn’t just mean being paid, it means that you’re in someway financially benefitting from doing the driving, which doesn’t just mean money. It can be as basic as being bought a drink.
Even in your car, accepting fuel money means you’re technically receiving a reward, and can invalidate normal car insurance.
Best option would be to contact the hire company and ask them, as I’m sure they’ll have specific requirements for any of their vehicles going abroad. Even if they don’t know the license requirements, any decent hire company will be a member of the BVRLA who should know the answer to these questions.