Is it possible to check if given exception_ptr instance owns an exception of certain type?

Suppose I have an std::exception_ptr instance. I need to check if underlying exception is of certain type (and ideally get access to it). It can be done this way:

std::exception_ptr p = ...;
try
{
    std::rethrow_exception(p);
}
catch(MyType const& x)
{
    ...
}
catch(...)
{
    // do smth else
}

Is there a better way?

Is there a guarantee std::rethrow_exception isn't going to copy underlying exception leading to problems like throwing std::bad_alloc instead?