Hi Lucas:
Allow me to chip in here as a Filmic Pro user who has experienced the same issue with out of sync audio once imported into Adobe Premiere. For your information, I am using an iPhone 6 and Premiere CS6.
Now, it is my understanding that all video from an iPhone is a variable frame rate. Therefore, since Filmic Pro is an iPhone app, there is no way to set it to a constant frame rate. Generally speaking, when you buy an actual camera and record in 24 fps or 30 fps, that is exactly what you are getting and that is exactly what Premiere Pro will see. On the other hand, with iPhone video footage, 24 fps is actually 24.02 and 30 fps is 30.02, and so on and so forth. This is where Premiere Pro has a hard time keeping things in sync.
Do note that most folks have no audio sync issues with Filmic Pro footage when using Final Cut Pro as their editing program. It seems like Premiere Pro simply does not like variable frame rates, never has and probably never will. Therefore, a solution for you could be to use Final Cut Pro instead of Adobe Premiere.
On a side note, do know that when you save your Filmic Pro footage to the camera roll of your iPhone it gets slightly compressed. The best practice to avoid any compression right from the get go is to plug your iPhone into your computer, go to iTunes and select your device, select ''apps'' and scroll down to the Filmic Pro app where you can then select your videos and save them to your computer without any compression. This is just a side note though, and obviously, you are free to continue importing your footage as you have been doing if you do not mind a little bit of compression from the get go.
Now, for the big question! How to turn the footage from Filmic Pro into a constant frame rate so that it is usable and not out of sync in Premiere Pro? The answer is Handbrake.
I know that you mentioned trying Handbrake only once, getting a huge file and that it was the wrong resolution to boot. Personally, I think the problem is that you gave up too soon and didn't bother to learn how to use Handbrake properly.
Using Handbrake:
1. You should import your file, or files, or folder containing the files.
2. In the column on the very right, under the heading ''regular'', you can select ''High Profile'' if you desire an H.264 codec level of 4.1 as opposed to the default H.264 codec level of 4.0. And very important, do not select a preset under 'Devices'. For example, if you select 'iphone & iPod touch', you will notice under the 'Picture' tab that the width of your video changes from 1920 to 960. That is probably how you ended up with a video that was not 16:9.
3. In the 'Picture' tab, there is nothing to change, leave it as is. Unless you want to crop your video or reduce the resolution of your video (i.e. from 1920 x 1080 to 1280 x 1080). But as I say, there is nothing to change here since those are things you would be doing in Premiere Pro anyways (cropping, exporting in lower resolution, etc.)
4. Generally, there is nothing to be changed under the 'Audio', 'Subtitles' or 'Chapters' tabs. Leave those as is.
5. The 'Video' tab is where you will want to be making changes, but only in one or two areas. A) You will want to select 'Constant Framerate' instead of 'Variable Framerate'. And B) If you want, you can play around with the 'Constant Quality' under the 'Quality' heading. Generally, you can leave this untouched at the default setting of Constant Quality: 20. There is no point in Constant Quality: 0 (Placebo Quality) as it will just increase your file size beyond belief without necessarily improving your video quality. My general recommendation is anything between 20 and 10 for Constant Quality. And, with this, you should actually end up with a video size that is a bit smaller than the original without actually losing any quality.
For example, my original 1920 x 1080 Filmic Pro video was 370MB. Once I ran it through Handbrake at Constant Quality: 20, it was 80MB. And, once I ran it through Handbrake at Constant Quality: 10, it was 114MB.
Also, the resolution stayed the same (i.e. 1920 x 1080 or 16:9) because I didn't make any changes under the 'Picture Tab' (size or cropping) and I didnt select a preset in the column on the right-hand side. Again, just leave things as they are. Handbrake will keep the same resolution as your source video unless you start selecting presets and changing sizes and stuff. Again, all you really have to do it select your file, change it from Variable to Constant under the 'Video' tab and export it (start button).
Now, a few notes: It seems like the audio sync issue is more severe or more prevalent when shooting longer video vs shorter video (i.e. 10 minutes vs. a 1 minute). Also, it seems more prevalent when shooting in 24fps as opposed to 30fps.
Hopefully, you can give Handbrake a proper try and have some success with your video editing of Filmic Pro footage in Premiere Pro. Feel free to let me know if you have any questions.
Cheers and best of luck.:)
Jade.