From the standard section 9.4.2:
If a static data member is of const integral or const enumeration
type, its declaration in the class definition can specify a
constant-initializer which shall be an integral constant expression.
In that case, the member can appear in integral constant expressions
within its scope. The member shall still be defined in a namespace
scope if it is used in the program and the namespace scope definition
shall not contain an initializer.
Basically, local classes have no linkage, and static data members require a linkage.
Since there’s no way to define a static data member of a local class in namespace scope (a declaration with an initializer is not a definition), they are not allowed, whether they are of const integral type or not. On the surface it may seem like the compiler should just be able to inline the value, but then what happens if you try to access a pointer to the member? With namespace scoped classes you’d just get a linker error, but local classes have no linkage.
I guess in theory they could just allow you to use static const integral types in local classes as long as they are only used in integral constant expressions, but it would probably just put too much of a burden on the standards body and compiler vendors to differentiate for very little practical value; local static variables are accessible from local classes, so using a local static const should be just as good.