What would be the difference between というのであれば and であれば?

というのであれば is a (stiff) conditional form of というのだ, which is と言う followed by an explanatory-の. In many cases, you can simply understood というのだ as "it is that one says ～". But when it is followed by から, (で)は, (で)あれば, か, etc., the literal meaning of 言う is often lost and the phrase works as an emphatic reason marker. See also: The usage of という in questions

In your first example, それは昔母に買ってもらったものであるというのであれば literally means "if it is that she says it is something her mother bought to her long ago", but you can treat this as an emphatic version of 'if'.

Your second example doesn't have enough context, but doesn't "if you say" or "if one says" make sense?