Given the small number of binary types available, and the fact that the name of each type spells out its size, defining such a function is trivial.
In[2]:= sizeMap = <|"Character8" -> 1, "Character16" -> 2,
"Integer8" -> 1, "Integer16" -> 2, "Integer24" -> 3,
"Integer32" -> 4, "Integer64" -> 8, "Integer128" -> 16,
"UnsignedInteger8" -> 1, "UnsignedInteger16" -> 2,
"UnsignedInteger24" -> 3, "UnsignedInteger32" -> 4,
"UnsignedInteger64" -> 8, "UnsignedInteger128" -> 16,
"Real32" -> 4, "Real64" -> 8, "Real128" -> 16, "Complex64" -> 8,
"Complex128" -> 16, "Complex256" -> 32|>;
In[3]:= sizeMap["Real32"]
Out[3]= 4