Use of part perfect continuous vs. Past Perfect

Solution 1:

I think you have misinterpreted what is needed to be continuous in this sentence. That is, the list of movies must be continuous.

This sentence only reflects on a finite number of movies in the past that were missed, but makes no indication that there might be more movies that she didn't miss:

She mentioned that she had missed several movies because of Paul's later arrivals.

had missed => that's the end of the discussion.

By using the past perfect continuous instead, the implication is that the action of missing these movies has ceased, with the expectation that there are definitely more movies and she has not missed those ones.

She mentioned that she had been missing several movies because of Paul's later arrivals.

had been missing => but now she isn't, or but later she wasn't.