This is how to achieve in-tile scrolling in Unreal.
I haven’t tested it in Unity, but the same logic should work.
Still the issue is pixel bleeding,
if pixel with color lies at the edge of the tile or too close to the edge.
While technically it’s possible to fix it by allowing some safe margin,
but it would be too fiddly to my taste and not worth the effort.
Another option is to use Nearest texture filter mode,
but the final texture will look blocky/pixel-art-ish.