I see. I have not implemented a custom projection before, only a provider and source. However, if you use a projection other than Mercator, you may not be able to see the base map (https://www.telerik.com/forums/custom-projection). Although that is an old post.
I do have a suggestion that seems a bit klunky, but might work. In your provider's GetTile or GetCachedTile, you can figure out the coordinates of the requested tile. Then convert/map those coordinates to your own tile. If your provider's tiles do not have the same positions (or size), then you would need to create a new bitmap using the portions of the tiles you need.
A good article on tile mapping to geography: