When Batman was airing on two nights a week, each 2nd episode
began with a recap of the previous night's episode. What I would
like to know is this:at point was the frame-by-frame recap
replaced by picking up where part 1 left off?

Eh, I'm trying to think about each and every episode. I'm thinking that the first two seasons did have the flashbacks to the first part. Season 3 would prob. come into my mind for the change. You had cliffhangers in a way. Let me explain this. Okay, the first two seasons the cliffhangers always had Batman and Robin in some sort of trap. You had to tune in to see if they lived or not. Season 3 had really cheap cliffhangers, and I really wouldn't consider them to be cliffhangers. Season 3 mostly consisted (if they were one parters) which crook would be in next weeks episode. If you like Penguin and the cliffhanger showed Penguin sneaking around in the library, you'd have to tune in to see what crime he would be committing.

I'm sorry, I can't really remember the exact episode they stopped using the frame by frame. I hope I cleared some of your question up. The next time I catch Batman, I'll look for the frame by frame and come back to this question.