OK @Aparecium I think I've got something!
It's called a Quick Part and it's not perfect, but may help - you would need to change picture for each incidence in your document, for example.
So, Insert your image, do the formatting, wrap, layout as you want it. Make sure to select 'Fix Position on Page' and deselect 'Move Object with Text'.
Once it's all where it should be, select the image, and go to the Insert menu bar.
Click 'Save Selection to Quick Part Gallery'
Give it a name. Choose which template it will apply to - important if you want this to be usable everywhere.
Now, this is where I haven't quite worked it all out fully, as there are options called Insert Content Only, Insert Content in Own Paragraph, or on a New Page.
I can't see the difference between 'content only' and 'paragraph'. Perhaps only relevant for text.
Anyway, whenever you want the image on your page, go to Insert, Quick Part, and the new ones are at the top.
If you need to edit the insertion style, it's Insert, Quick Parts, Building Block Organiser.
Personally, I would make several Quick Parts if you want more than one image on a page, because it does literally overlay the original. It's retained every single piece of formatting.
Thank you for the project! Even if it's not helpful to you, it's given me something new to learn.