And a P.S. on entrance size: it can be very small, for a small hive. 1/2 in high, one inch long still lets 2 or 3 bees at a time pass. That's plenty when there's no nectar flow.
Assuming they are not diseased, I would baby them along, unless they are queenless. Classic technique is to first give them a frame of emerging or capped brood from another hive. Eggs/larvae may be too much of a burden on them. Even if you can spare no brood frames, adding cup of nurse bees from a stronger hive can help.
Another help is temperature regulation. If it's all they can do to cool the hive, then they don't have enough workforce to expand the colony. When a beekeeper can help them stay close to their optimum temp, they build faster. Open screens may let in too much hot air...it's been said, like running the a/c with the door open.
You may also want to research keeping small colonies condensed with a follower board, making sure they don't have too much space to regulate, a/k/a "tight bees."