Red Bee,
I think you are looking at spliting your hives, not out-and-out queen rearing. I'm guessing here that you are just in your second season at present?
Queen rearing is when you set about raising several queens in a batch and likely followed by more batches. With just three colonies, you have little chance of doing that. Finman is right in what he says. Your 15 queens ('rearing') would be costing you, in all your potential honey crop, which from 3 colonies can be considerable.
Even if only 75kgs (and could be much more) that may raise over three hundred pounds worth of honey to help build/buy some extra kit. Your 15 queens may only make a dozen going into winter, what with the odd queens being just poor, of nasty temperament, weak colonies - leading to losses to wasps and robbing. With no backup from which to draw reinforcements later in the year, you actually run the risk of making very little progress in a bad season. That can be very disheartening, even more so if there is a large sugar bill and no honey either!
Much better to do as you suggested and settle for strong splits by artificial swarming, (with a few extra splits as you might be able to manage), rather than trying queen rearing. When you get to double your hive count, you might be having a tentative start to queen rearing; ten colonies would be better.
The 75 boxes and 750 frames is just being a bit OTT. They would not all be needed next year - but likely for the following, though!
Regards, RAB