If Silent Lab stadium is in play and one of my Basic Pokemon evolves into a BREAK Pokemon, would it be able to use the Ability from the underlying Basic Pokemon or not?
When it evolves into a BREAK Pokemon it is no longer a Basic Pokemon, so Silent Lab no longer affects it and you would be able to use the underlying Basic Pokemon's Abilities.
Source: TPCi Rules Team (2016-06-23)

Relevant cards

Note: All applicable cards may not be listed below.
XY5_140