Is Cu(OH)2 soluble in an excess of NaOH ?
It is more soluble in NaOH than in water, but it is a 'meta-stable' solution.
It's solubility increases with increasing NaOH up to a maximum ratio, after which it falls off sharply again.
See the chart in the following reference:
A log K of 9.36 ± 0.02 was most consistent with the experimental data for Cu(OH)2 solubility,
Loay Hidmi and Marc Edwards, Environ. Sci. Technol., 1999, 33 (15), pp 2607–2610, June 26, 1999.