Even if I make the text invisible it still creates an extra page in the ebook for the text.
Two tweaks: make the anchor paragraph leading 0, and anchor the image to the END of the anchor paragraph. With that, making the image appear on the same page becomes entirely a matter of the image size in the EPUB reader. If it is full page height, leaving not even that smidge of room for the anchor paragraph, one or the other will break to the next page.
To be safe, I'd keep the Part image at about 2/3 page height, relative to the margin width. That should leave enough slack that all readers keep it on the actual anchor paragraph page.
You will want to experiment with the image sizing options in the Object Export Options menu (right click on the image frame), to override the global option set at export. You should be able to get a full-width, maximum height Part image with one or another setting there... but keep in mind that the result will be reader, and user-settings, dependent.