How can I upload high-res photos to Facebook from my iPad

Probably the best way is to do it through Safari. As far as I know Safari doesn't change the size or quality of uploaded images.

Keep in mind that max size in pixels of an image on Facebook currently is limited to 2048x2048px. If you try to upload a bigger image Facebook will resize and reencode it automatically and the result will not be optimal.

I uploaded a photo through Facebook, but it had bad quality, so I tried uploading it with iPhoto, and it did it with the highest quality. I also compared a the quality difference between uploading it with the FaceBook app and the iPad version of iPhoto and it too looked better!

I think that iPhoto would not give you a choice of photo resolution. Just upload the photos through the Facebook app for iPad.

Here is official Facebook answer to your question: https://www.facebook.com/help/266520536764594

You may also consider this option while creating FB album: https://www.facebook.com/help/187741037945488

Also, if you use Facebook iPad App to upload photos, make sure you have 'Upload HD' option turned on in Facebook iPad App -> Menu (upper left corner) -> Settings -> Videos and Photos -> Upload HD.

You may use any application on iPad (for instance, Snapseed, which is good), however all of them will meet same limitations caused by Facebook policies mentioned above.