Substitute for butter in a burger recipe?

Personally, I would omit the butter and switch beef cuts from the leaner sirloin tip to something with more fat, like chuck roast (shoulder).

Looked at the recipe...normally I use olive oil or a good cold pressed oil when I want to replace butter (volulme for volume) in a meat recipe (made at room temp).
But looking at your link (where the butter is distributed in little chunks amid the ground burger) and thinking that butter is part water, a straight oil substitution might be too greasy and release too soon.
What I'd want is a fat that is solid at room or slight chilled fridge temp.
Coconut oil and schmaltz (chicken or duck) are possibilities, but each would have its own flavor. Maybe too distinctive.
A related idea, what about getting some kosher beef fat from your butcher and using that?

I put a little olive oil in my turkey burgers. . If you are using 80/20 chop meat I don't even think you would need the butter. You are really just adding more fat into the burger.

I'd use frozen bone marrow. That should add the richness and flavour that the butter does. It would add another step, as you have to first release the bone marrow from the bones (I soak them for a few minutes in tepid water. The marrow then pops right out), but the rest of the stages should be the same.