Delighted you've got such a good result, OP. I hope your daughter enjoys every second.
I'm a little stunned at the response from the ROH. I realise it's huge and must have a rather 'computer says no' type attitude in order to function but, given that they responded to an email about a social media thread, I am actually really surprised they were so negative. A performance at the ROH can't be that common a 'wish' so I doubt it's a request they get all the time.
I don't think it's that they're worried there will be a slew of people wanting their under-fives to see a regular performance, more an acknowledgement that a regular ROH performance, outside of specific family-friendly events, just isn't a suitable environment for a three year old. Performances are extremely formal, the audience is numerically dominated by older people, and the expensive seats are likely to be occupied by longterm sponsors of productions/renovations/Young artist training programmes they are desperate to keep sweet and who wouldn't find a three year old dancing in the aisle at all charming. General rehearsals are serious, working affairs, with half the auditorium taken up with lighting desks/technical stuff, photographers taking publicity stills and video, the directorial team poring over scores, taking notes and consulting with the conductor in the pit etc etc.
I have a well-behaved, very musical five year old I could technically bring, but it really isn't the place for him despite the fact that he would be quieter than the deafening coughers who specialise in sounding off at quiet moments in the score-.