Do you want to pay on the front end or on the back end?
Pay for queens now, or pay for lack of production by making the bees make their own queens.
Yes, you run a risk of the bees swarming by limiting space and inducing them to swarm. With added risk comes potential rewards. With the added risk comes the potential reward of being able to split more ways.
I was hoping to keep the queen I have in the strong hive, so I can pull frames of brood to strengthen other hives.
Multiple mated laying queens will produce a lot more brood than one queen can.
Could you possibly do a cutout of some cells (I'm thinking a small circle cookie cutter maybe) and then placing them in a frame (with circles cutout) and put that in a queenless nuc? I'd wait for them to make the queen cells, and then transfer them to another nuc. Maybe I could put the cut-out of cells in a wooden frame so it would be easier to move, and damage the queen cell less.
The two most difficult things in beekeeping are producing comb honey and raising queens.
Anything is possible. Sure sounds like a lot of hard work to me. Your time might be better spent picking up pop cans for recycling money, and buying mated queens.
Yes, you can make a split and let the bees raise their own queen. However, this comes at the cost that you are unlikely to be able to split them again and have the splits be strong enough to overwinter. Plan on the bees needing about 3 months after a walkaway split to build up enough strength to split again. Plan on being able to split again in 5 weeks if you add a mated queen.