The reason you can't do this is because non-constant expressions can't be parsed and substituted during compile-time. They could change during runtime, which would require the generation of a new template during runtime, which isn't possible because templates are a compile-time concept.
Here's what the standard allows for non-type template parameters (14.1 [temp.param] p4):
A non-type template-parameter shall have one of the following (optionally cv-qualified) types:
integral or enumeration type,
pointer to object or pointer to function,
lvalue reference to object or lvalue reference to function,
A non-type template argument provided within a template argument list is an expression whose value can be determined at compile time. Such arguments must be:
constant expressions, addresses of
functions or objects with external
linkage, or addresses of static class
members.
Also, string literals are objects with internal linkage, so you can't use them as template arguments. You cannot use a global pointer, either. Floating-point literals are not allowed, given the obvious possibility of rounding-off errors.
Now an impl would need to come up with a unique sequence of characters for a std::string or, for that matter, any other arbitrary user defined class, storing a particular value, the meaning of which is not known to the implementation. And in addition, the value of arbitrary class objects can't be calculated at compile time.
It's planned to consider allowing literal class types as template parameter types for post-C++0x (see below), which are initialized by constant expressions. Those could be mangled by having the data members recursively mangled according to their values (for base classes, for example we can apply depth-first, left-to-right traversal). But it's definitely not going to work for arbitrary classes.
As of C++20, we are now allowed to use structural class types as template parameters. In a nutshell, structural classes must have a constexpr constructor, destructor and only structural-type members and base classes (like scalars, arrays thereof or references). They must also only have public and non-mutable base classes and members. These provisions, if the template is instantiated with a constant expression converted to the parameter type, allow the compiler to mangle the argument meaningfully.