If this barberry is inside, that's a primary problem. The fuzz is mold probably growing from dead buds at the base of the leaves. That mold is NOT killing your plant. It is only cleaning up what is dead and dying -as in decomposing.
Barberry won't live inside -- it may limp along for a considerable period, since it's a tough plant to begin with, but it will eventually croak from dry air, too much water and not enough light. What you're seeing is the final stages of its life as the roots rot and leaves drop. The new growth is a last-ditch effort the plant is making to survive. This may sound dire, it is.
The soil doesn't drain and you're probably watering it regularly or constantly, contributing to the root rot that is a result of too much water.
To save this will require an emergency repot into better adequate bonsai soil and a complete root cleanse to remove dead roots. After its repot, it should be placed outside in full morning sun where it will get at least four hours of sunshine.
You don't provide your location, which is critically important for anyone here to help you...
