AVCHD is not one of video formats that Final Cut and iMovie can edit directly,so your AVCHD can not be recognised by Final Cut and iMovie.
The video formats that Final Cut and iMovie can support are MPEG 4, MOV and H.264.So you should convert the your AVCHD(.mts) video to video formats that compatible with Final Cut and iMovie and then do some editing via them.
As for MTS Converter, i recommend Aunsoft MTS Converter for Mac which is professional and powerful in converting MTS files to various playable and editable video formats, like MOV,WMV,MPEG,MKV,AVI,FLV,MP3,FLAC,OGG and ect.
Additionally, this software provides some editing functions like cropping video black edges, trimming video length,adding image or video to original video.I like this function, because i can make my own movie with this software.
Enjoy your videos.