There are two ways to edit AVCHD video.
You found the first method. Copy the MTS files from the camcorder to the computer and "piece them together". Your observation of the 2 gig file size limitation is known and is basically required by certain operating systems which cannot deal with large file sizes. Many times, this method requires transcoding (converting) the copied MTS files into a more useful file type and uncompressed format for the editor to deal with.
The other method generally means connecting the camcorder to the computer with a USB cable, putting the camcorder into "Play" or Computer mode and doing exactly what you say you can't do - but this does assume that the video editor is AVCHD-file capable. During that "import" (capture) process, the camcorder's electronics are used to decompress the video file data stream and the video editor can pull the video into whatever video type it likes to deal with. Sony Vegas and Apple iMovie and Final Cut can do this - I was pretty sure Premiere could too - though not know which version you are using makes it challenging. When the video is imported/captured this way into the computer, the "seamless" video you want should happen.
The problem you are encountering with the method you are using probably has to do with the way AVCHD video compression happens.
Frame 1 = base
Frame 2 = what changed from frame 1
Frame 3 = what changed from frame 2
Frame 4 = what changed from frame 3
Frame 5 = what changed from frame 4
Frame 6 = what changed from frame 5
Frame 7 = what changed from frame 6
Frame 8 = what changed from frame 7
Frame 9 = base
Frame 10 = what changed from frame 9
Frame 11 = what changed from frame 10
Frame 12 = what changed from frame 11
Frame 13 = what changed from frame 12
Frame 14 = what changed from frame 13
Frame 15 = what changed from frame 14
Frame 16 = what changed from frame 15
Frame 17 = what changed from frame 16
Frame 18 = base... and so on...
If the file size limitation happens at frame 4, then frames 5, 6, 7 get confused. If the second method (importing/capturing) is used, then the camera's file system working with the editor are able to piece the video back together as a contiguous file and the "dropped frame" issue should go away.
If you continue to use option1, then be sure to use the highest quality capture setting in the HF M31. This way, there is less compression of the captured video so more of the frame is available following the base frame. Because of the reduced compression, there will be more of the segmented files.