To the original poster.
There are no circumstance with a conventional aircraft, where adding even little flap would allow to make the runway which was undershot in clean configuration. I might be wrong, but it seems to me that it is even mentioned in basic Principles Of Flight ground school subject. This is due to the fact that the wings for most aircraft is designed to have the best L/D ratio at the clean configuration, and that is when the aircraft will glide the farthest. Having flaps extended, improves lift, but also changes the drag, and it does not give the same maximum L/D ratio any more, thus the glide steepens. It is possible that there are some airplane designs which incorporate the improvement of L/D ratio when flaps extended, but I suspect it is very hard to land such aircraft, because it will tend to fly at the point when it should stop flying.
To assure myself, I have event taken a Stick And Rudder copy from the shelf, and it's clearly states "Flaps have a double purpose of lowering the airplane's stalling speed and of increasing its drag, thus steepening its glide. As far as glide control is concerned, only this drag effect is important. With flaps down, the airplane's glide is so exceedingly steep that there is hardly any problem of glide control;". Not all are fans of the book, but it just proves that the basic principles exist more than half of century, and I wonder why would anyone suggest otherwise, except proven in practice.
As per the whole situation. I think You had a very good opportunity to learn many things and make conclusions. Your initial feeling was correct, and the Go-Around would have been the right action (You mentioned that You'd have done it if being alone in the cockpit, or with an examiner. I don't think this decision would count as a fail - it is good judgment). The tricky part here is that You were flying with instructor, who is the PIC, and in general, You have to trust him, because he is supposed to "know better". Depending on how long You are in training and how much experience You already have there is a certain degree of psychological influence on the student's decisions because instructor is supposed to mentor/guide the student and demonstrate things. So You got in to the trap of following suggestion of Your instructor, which is the right thing on it's own. You cannot blame Yourself for this. The issue probably was that instructor chose incorrect strategy of demonstrating his point. The ideal course of actions in that situation, IMO, could have been "You call Go-Around and execute it. Instructor accepts it, but tells that the approach could have been executed. On the next circuit instructor takes the controls and demonstrates from the same aircraft position. The exercise is finalized by the 3rd circuit, where You execute it Yourself". This didn't happen, but it doesn't mean it is Your fault in this case. Maybe next time You should call for GA as soon as You feel it and not hesitate. Don't blame Yourself and don't blame anyone, instead learn from it. As Your experience grows, You will realize that instructor wasn't always correct, but it's ok, we are all humans and all make mistakes. You are lucky to realize this now. Since You have posted this situation, and there was a discussion on it, I believe You have debriefed it quite extensively, so next time You will make the right decision.
Don't feel unsettled. This is a good lesson for You, not all have the luck to experience it during the training.
Good luck on Your further training.