Even in the best season, raising a queen in a half-populated nuc is tricky proposition, the little cluster simple doesn't have the resources to devote to raising a quality queen. The "give-them-an-egg" strategy works best in the height of the season with an overflowing single.
You could take the existing queen and some frames, and let the residual hive struggle with the raising-mating-restarting brood lottery.
This is a lesson I must learn myself every year. In April and May (the height of my season), queenless splits to 2x stacked 5 frame Medium nucs **all** take and boom quickly. I decide I have the magic touch and extend the queenless splits to nucs made up in July. My take is perhaps 50%. My July (a dearth) is like well watered area's September. The half of the July nucs that do take often struggle as dinks and go through supersedure cycles, as if the queen is an incompetent layer.
I have an Ulster style nuc top (can be strapped to any of my standard nucs). Just this week I put a frame on young larvae in the observation portion, and watched as the hungry bees devoured the brood for nutrition.