It looks like the highlighted bits of the field background are supposed to be the edges of "extra objects" on the field (i.e., objects that you can walk "behind" in the game, objects that have an animation, ...). They're not supposed to be transparent, but they're supposed to be displayed at their natural resolution with no filtering and thus have "clean edges."
Actually, this is the result of transparent pixels being blended with non-transparent pixels. Whether alpha channels are supported or not (varying degrees of transparency), the result would likely be similar.
In situation 1, where there are no alpha channels, merely a transparent color within a pallete, the transparent color becomes blended with opaque pixels, and the result is what you see above. More opaque, but extremely off color pixels.
In situation 2, where textures are utilizing alpha channels to determine transparencies, we might gain acceptable results, but not if there are any background pixels behind forground pixels. Essentially, pixels which originally should have been 100% opaque, become blended with pixels that were supposed to be 100% transparent, generating some inbetween results. If the back-backgrounds where a complete image, this would be little issue, but if forground images are literally cut out of the backgrounds, we get borders of n% transparent pixels that look through to... well, nothing, really.
There isn't much of a reasonable result to solve this problem, unless we're looking at post processing effects. Applying filters per texture just doesn't produce nice results in FF7. Post processed filters applied uniformly to the entire output tend to have a better, albeit, lower quality effect. Not to mention may generate artifacts of their own.
Anyway, to be honest, I'm not very fond of the bilinear filter. Far too blury. I hate to say it, but I personally feel that if backgrounds are going to be improved, it will have to be done manually, with some heavy hacking. In short, if the most dating factor of FF7 is going to be fixed, we might as well just write an entirely new engine, since overhauling each background individually is going to be more than enough work to merit the additional effort.
Not to say the effort is waisted, The SaiNt. Hacking FF7 to run in higher resolutions is by far one of the greatest improvements to the game to me. From here out, any additional work are bonuses.
