I think 3 is far too young to take a child to a full length piece of theatre, for which other audience members have paid a LOT of money. See if there's a film broadcast - but even then, it's long, formal, and again, the cost of cinema tickets for these live broadcasts can be up to £20.
If your child is likely to be restless, kick seat in front (even by accident because she's restless), wants to whisper to you, be taken to the loo in the middle of an act, etc etc do not take her to any of the large company productions (Birmingham Royal Ballet and English National Ballet tour, but I don't think they have a Swan Lake in this season's touring repertoire.
Even if you find a local amateur production, it's a long piece, with a lot of quite formal classical stuff. And the perfrmers and audience are no less worthy of being watched and performing in the same conditions as those at the ROH. As for "letting your daughter dance in the aisles" - words fail.
so -- Why not look out for a ballet specifically for children? At 3, she won't be doing proper ballet yet - just "good toes, naughty toes" stuff, so I'd think that a live engagement with a ballet designed for her age group and often including the possibility of an accompanying workshop.
I don't know where you are but what about Northern Ballet ?
Ballet Black was touring a show for children last season - you may find they're coming near to you soon.
If you find a performance that's doable, ring the theatre & ask if there are any accompanying education events: usually companies make much-needed extra funding from offering educational workshops, meet the dancers, etc. All that will make a FAR better introduction to the live theatre than sitting through 3 hours of a big classical ballet, with conventions that many adults don't understand, let alone a 3 year old.
I've been raised in the theatre (on stage in utero!) but we never took our children to the theatre until they were around 7 or 8. It's not fair to the kids or the rst of the audience.