If it’s moving with a chain tug then it’s the chain tug not setup correctly. No matter how you clamp it, I admit some clamping devices do a much better job of “clamping”, the wheel cannot move forwards because the tug is up against a immovable object, the frame.
As others have and I could offer better solutions, but you’ve got the frame you have and need a decent chain tug. The Surly ones are IME the best. When setting up make sure the chain tug is pulling the wheel back to tension the chain, then clamp up, then double check the tensioner is still sitting against the frame and the tensioning screws are still “tensioning” against the dropout.
If that doesn’t solve it then I’d look at the chain ring. Is it properly tight and running concentric? Do you tight and loose spots on the chain tension when you turn the cranks? If so the chain ring might be running out out concentric and or your chain might be over stretched in certain places.
Not related but what chainring and sprocket sizes are you using?