Re: TTA and Stagebox - we are with neither but have close contacts with each of those. Both run audition days and you sign up for that via a form on their websites.
Stagebox options are musical theatre (which would be a triple threat situation so dance expected as far as I know) or an all acting option which would be for screen. Everyone I know personally with Stagebox joined via one of their audition says but that doesn’t mean they don’t open their books at other times of the year. There is a fee for the audition day but it’s run like a workshop so hopefully she would feel she got her money’s worth. Stagebox also do training so I would make sure you’re up for their training options if you’re considering them as an agency. You audition to get a training place and then they talk representation after that.
TTA have auditions next month I think. They represent kids who train at a variety of places in a range of disciplines, so she might be more at home there without dance.
She’s too old for children’s West End roles and like others said, there’s then a gap - there are next to no stage roles between 12 and 16. A focus on screen work would be proactive if it’s something she would like. Voiceover work could also be good for her as there are animated roles that require singing as well as acting.
Back to your original questions: there are rubbish scam agencies and then there are very many reputable agencies and there’s not really a best place to go - it’s about finding a good match for you. Sometimes due to communication styles - some more blunt, some not; some phone, some do everything by email and expect you to always be checking your inbox. But also because you don’t really want to be too similar to another kid already on their books. If there is another girl the same height, similar look, similar talent, they may end up having to compete against each other in a way just to be submitted for a role, so it’s handy to avoid that. Keep in mind if an agent asks for money up front aside from booking a headshot session or a workshop day, then I would move on. Agents make their money by taking their cut when they are successful in getting you work and shouldn’t be trying to earn money from signing kids to their books.
Agencies attached to training have pros and cons. Pro being that the kid is constantly in front of the agents and hard work and progress and talent will all be noticed and keep you on their radar. Con being that you may find they offer more and more classes (costing you significantly more) and it can be hard to say no when it feels like it will help your child’s chances. It can help to set a budget before you get to that situation and then if more is offered, say you can’t afford this as well as that; which one option would be more beneficial, etc. Do see what their opinions are on studying with teachers outside their organisation as well. Imo agencies that disallow study from anywhere else are not in the child’s best interest. (For context, I have a professionally working child and our agent is through a theatre school and we are very happy there, but we do also train elsewhere and we don’t do any secret keeping about that!) It sounds like the opportunities for young people at the Royal Opera House might be right up your daughter’s street and you wouldn’t want an agent to talk you out of that in favour of something inferior.
If you haven’t already, I would make sure your DD has a good current headshot and a self tape with a monologue and a song. Having that ready will make approaching agencies much easier.
If you’ve read all this, my last thought would just be to make sure this is the route your child wants to take and that she’s cool with missing social opportunities with friends, dropping everything to tape, training a fair whack but still keeping schoolwork up to date (school need to be on side for licensing). And you need to be supportive enough to facilitate self tapes and flexible enough to drop everything to take her to castings and/or work. Saying to an agent that you can’t make something will hurt her chances of being submitted in the future, so it’s good to go into it with open eyes.
all the best for your DD finding happiness and success!