I have a lot experience what to do with little hives. 2 pounds (1 kg) is the smallest unit which is worth totry to rise upp to colony. It takes 2 month to get it to gather honey.
After winter I have that size of colonies.
Also if I make a nuc for queen mating and I make a honey colony for next summer.
First I limit the space of box that frames are full of bees. No extra space or frames. Colony will deminish untill new bees began to hatch after 3 weeks.
Now I have a system that I can speed upp 3 fold the development of little colony. I heat the box with terrarium heater and I feed them with pollen+yeast + soya.
When you have heating 8-15 W bees can spread over the langstroh box.
The number of nurser bees limits tha brood area.
When new bees hatch, hive has enough nurser bees. Now flying bees are at their minimum. It takes 3 weeks after new bees are able to collect pollen and honey enough from field. So 1,5 months has gone.
If queen is good, it fills the hive with brood after 1,5 month after istalling. It has 2 box young brood, lets say, in the middle of June. That kind of hive is able to collect honey normally at the beginning of August.
I suppose that if hive is now in the middle of May 5 frames, it will be ready to collect honey after the middle of July.
You have warmer climate but yor have colder uninsulated hives in the south. Night temperature limits the growth of brood area.
I make my terrarium heater experiments in August. I clearly saw that temperature limits the brood area. During that nuc rising I have no idea to feed sugar syrup. But I gived protein patty, which has 50% sugar.
In summary feeding all the times with sugar syrup is not good for bees. It limits the space of brood area. But I have seen that many give syrup also for 2 box hives. In Finland that it is forbidden. There cannot be cane sugar in sold honey.
When I got a swarm I give during one week 20% sugar syrup. Bees build the combs during a week and sugar is then used.
If I give stronger syrup they cap it in fames. It limits the brood space.