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Lerping across vertices

I have a Runtime Mesh Component, and I use UV to pick a texture from a combined file for each tile. What I want to do now, is lerp (at the edges of each tile) the texture with the neighbouring texture, so that I get a smooth transition.

How would I go about doing this?

The RMC construction script:
593f1c9bd57d4fdc4768ce4d8ac00717aec9bfe8.jpeg

Result:
4a3e10fa32cb5247c1eabdd69b336112683eecf3.jpeg