It's a balance isn't it.
I had a job at 16 and that gave me more financial independence to spend on silly adventures with my friends (holiday to Newquay, cinema, nights out out (different times!), nail varnish, magazines etc but mainly in hindsight it was good for me to have experience from holidays and weekends of working in offices, shops, restaurants, bars.
My parents paid for my essentials and the normal stuff. I didn't do any summer schools or courses or paid activities as we couldn't afford it but some friends did national youth orchestra type things.
That's what helped me get my foot in the door at my first real job post uni - the holiday office work which meant I was the only applicant who'd used a certain accounting package at my age.
The customer facing stuff is valuable too, it's good practice for all of us to have real understanding of how challenging it can be and to flex those social skill muscles and problem solving.
Of course I loved also having plenty of spare time to sunbathe while listening to my favourite CDs, or reading books in my bedroom, or playing badminton or frisbee at the park while not so secretly watching the hot older boys play football. BUT it does no harm to have some structure and paid employment.
My older son has taken up all opportunities to work as he wants and needs the money. He does still moan about early mornings in the holidays but he's buzzing when he comes home as the lads he works with are great with him.
I think you and your daughter need to find a happy middle ground - is this office job full time? Would a 2-3 day a week option be enough? You could agree that you're paying for the dance course but that she won't get any other pocket money or handouts for going to the cinema or the beach or a curry. That's the money that she needs to be working for.
Final thing - I see so many people locally on facebook who are desperate to find a job as a 16 year old and it's hard unless it's a family friend's business. I don't think she realises how good she's got it!