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The camera footage shouldn't have an issues in AE, if you're using a registered version and not a trial. I use Sony MTS files all the time.
As far as converting goes, you have to make sure that all the settings are the same as far as field order, resolution, pixel aspect ratio, etc. when you're converting. Your description of your resulting files sounds like some, if not all, of the settings I mentioned were wrong.
Actually, the AE should recognize the AVCHD footage, since I imported one of my recoedings, which is 1080p MTS to AE, and it worked. Maybe the problem lies in 1080i. I always deinterlace my 1080i files and then import them to AE. You may try to deinterlace your files and then import them to AE.
As for the deinterlacing, you can google or you can give brorsoft mts converter, which I'm using now, a try. It has a codec for AE and Adobe Premiere, wchich make it easier to import files to AE. Besides, you can convert your 1080i files to MPEG-2 and then import them to AE.