There are many factors involved in your question and I am not experienced enough to make suggestions. The simple answer is that the best time to split is when the bees are ready. I split last year on April 14th because they had superseded and the hive was queenless. I ordered a queen for the big hive and put all the cells in a new box. Splitting definitely slowed them down but I still got honey and now I have two hives. My goal is to increase with my own stock. This year I want to start some queen cells and use those to make mating nucs.
In our area the middle of April is hard to beat. If Introducing queens I like to make them up the evening before and put the queen in the next day. I try to make them up using sealed brood. I have had them make emergency cells when using open brood.
If letting them make their own queens, I have had good luck putting the queen from the hive in the nuc and letting the hive raise the queen. If you have the resources, you can take some of the queen cells and make up other nucs as well. Sets the Hive back some though. Hard to make bees and honey.
Too many variables in your O.P. to provide a simple answer.
The state of the colony being split, in combination with the status of forage availability and weather are the major factors I look at when deciding if and how to split a colony. As for how to deal with the queen issue, that is also open to options - since I raise many queens, and most of the season have queen cells available, most often I would give the queenless splits, ripe cells of my chosen MQ (mother queen), making sure that I use cell protectors if there's a chance the resident bees might abort the planted cells. If the genetics of the original queen are questionable, most likely I would also replace her with a ripe queen cell, as soon as it was convenient.
Weather and forage (or ability to feed), would be primary factors in how viable smaller splits might be. I often like to make as many splits as possible, from colonies chosen to be donors for split production.
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