template<class _Ty, class... _Types, enable_if_t<!is_array_v<_Ty>, int> = 0>
inline unique_ptr<_Ty> make_unique(_Types&&... _Args)
{
// make a unique_ptr
return (unique_ptr<_Ty>(new _Ty(_STD forward<_Types>(_Args)...)));
}

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since we don't know which member functions the template is going to check

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plus any templates is also calls... Which is why I am wary about this.

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class felineTestConstructorOverloads
{
public:
	felineTestConstructorOverloads(long nSizeOne, long nSizeTwo, long nSizeThree);
	felineTestConstructorOverloads(char *pszName, long nSizeOne, long nSizeTwo, long nSizeThree);
	int getVolume();
};

void testingTemplateTypeParameters()
{
	std::unique_ptr<felineTestConstructorOverloads> uniqueFelineTest =
                std::make_unique<felineTestConstructorOverloads>();
}

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So you are not really asking about the parameters of the template class, you are after information on the type you are using with the template, in this case your class "A"?

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For other template classes though, the restriction might not be on the constructor. Or there might be more than one restriction.

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Trying to summarise all of this in a tooltip could well run away with us, but here VA View should help quite a bit, so long as you have some idea of what the template type is doing with the type it "contains".

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template<class T, class... TArgs>
unique_ptr<T> make_unique(TArgs... args)
{
    return unique_ptr<T>(new T(args...));
}

class A
{
public:
    A(int i) {}
    A(int i, float f) {}
};

std::unique_ptr<A> ptr = std::make_unique<A, int>(1);
// or
std::unique_ptr<A> ptr = std::make_unique<A, int, float>(1, 2.3);