Unfortunately, the character images have that grey transparent layer as it's part of the game, which I was disappointed in too. The characters in the pre-reg should have transparent PNGs so you'll probably have to go their individual character pages in the English wiki and save the character renders. Don't bother searching the Japanese wikis as they either have the same results you have or terribly rendered out character images from the game's stock PNG files.
DMM KamiPro's Twitter account had some PNG images in their old posts for Amon, Lancelot, Belobog, Dike, and Michael. I only rendered out Amon's image from the black background so someone will probably have to render out the others.
After one week of searching in Google images (yeah I resorted to that), I found a site through archive.org that had JPG character images of all the current characters as well as some of the newer ones. Unfortunately the JPG images are not in the highest quality, but hey it's better than the game's small character images. The new characters' images are HUGE though.
Here's my album with their character images. If you want, I can get the rest of the Eidolons. Only have all the Kamihimes and SSR Eidolons available at that site so far. The order goes JPG > PNG renders > PNG Twitter images.
Kamihime Project Character Images