I wouldn’t say lying that you have an ex that you don’t actually have is a “big lie.” I mean yeah its a lie he shouldn’t have made, but I wouldn’t call it a big lie or even an important one. It does set a precedence of lying, which is bad, but the lie itself is one that really holds no weight on tge relationship (besides the act of lying in of itself). However the partner is not beyond fault either. Being this pushy about an ex is (I feel) a red flag. In the end, both parties involved are being stupid and toxic and probably just shouldn’t be together in the first place cause it will just escalate unless they can grow enough as people.