The BPM Counter is too rudimentary for this kind of work. Beat mapping is the only way to go here.
I will strongly suggest that you save a copy of your project which has ONLY the imported soundtrack from the movie file, with everything in sync as you described. For each individual scene or cue, do as "save as..." of the master project, i.e...
- Master Project
- Master Project, saved as "1fx1"
- Master Projects, saved as "1fx2"
...etc...
When you make these copies, save each one with the save assets box [i]unchecked[/i].
When it comes to beat mapping, Logic does a fantastic job. But I would strongly suggest that you do not attempt to work on two or more scenes in the same project.
Finally, workflow considerations...
1) For each scene/cue, load up the Master. Assuming you want each scene to start near the beginning of the project (1 1 1 1), determine where you want the movie to start by regular playback (or otherwise positioning the playhead to around the start of the scene). Note that SMPTE number. Then, SMPTE-unlock the audio track and move the left edge towards the right to past the point where you actually want the movie to begin playing at 1 1 1 1. (Alternatively, use the scissor tool to slice the audio a bit past the point where you want the movie to start and discard the portion on the left). SMPTE-lock the audio again. Then [i]set[/i] the SMPTE Position of the first tempo event (in the tempo list) to "read into" the movie so that your scene starts in the vicinity of 1 1 1 1. Finally, SMPTE-unlock the audio track and drag the left the left edge of the region to 1 1 1 1. Then Save As... (as described above).
With respect to the audio track, the operation I described above is a necessary evil in Logic.
2) Do beat mapping first. Be sure you're satisfied with the beat mapping before doing any other work
3) Many times it's unnecessary to beat map the entire song. You can beat map strategic downbeats and get the tempo of the whole song from there. If the original song was recorded to a click and doesn't change in tempo, chances are you can beat map the downbeat of bar 1, 2, and 3 to establish the tempo for the rest of the song. Just to be sure, add another beat mapping 'node' at the downbeat of the last bar of music. Then delete any 'nodes' between the downbeat and the last beat. That will give you the average tempo for the entire song.
4) Do not worry in the least if your tempo ends up looking "fractional" (per your post). It doesn't matter what the numbers look like. The point is to get a steady click.
HTH
