I have looked in to this briefly before. Generally the bigger the core barrel the better your recovery, which means better samples for tests. Although on the flip side, Hoek and Brown show that UCS decreases with increasing core diameter.
Based on International Society of Rock Mechanics (ISRM) Suggested Methods (ISRM 2007), the diameter of the core specimen used for uniaxial compression testing should not be less than 54 mm, while it is 47 mm for the American Society for Testing and Materials (ASTM; ASTM 1994; ASTM 2000).
NQ = 47mm dia so still acceptable to run UCS test. Note sure on any other limitations.
We typically use PQ which is 85mm dia core. Sometimes we have to change to NQ to run pressure meter tests. Have never been questioned on the core size possibly leading to lower UCS. I think clients might be more concerned with quality of rock recovered that a slight loss of strength as fractures, discontinuities have likely a bigger effect.