The caterpillar of the Polyphemus Moth can eat 86,000 times its weight at emergence in a little less than two months.
Your birch tree will be not enough.
I agree that this is a Polyphemus Caterpillar, Antheraea polyphemus. But I think you exaggerate about it eating the whole tree. They barely weigh anything the first 2 or 4 weeks and at that age one still can't consume an entire Oak leaf in a day. The one pictured looks older than that and can probably eat a few Oak leaves (as an example plant) in a day but nothing to noticeable. By the end of it's life cycle it may have stripped a small branch of it's leaves but over a two month period that's nothing the tree can't simply regrow.
Again if Mothra seems to be eating enough to make the tree unsightly I urge relocating it to another host plant. Use a stick or tongs, or snap off the part of the tree it's on and put it in a container. Host Plants include: Apple, Ash, Birch, Dogwood, Elm, Hazel, Hickory, Maple, Oak, Rose, and Willow, with Birch, Rose, and Willow at the top of their menu. I emphasis not touching them with your hands as the hairs can be irritating and sometimes cause allergic reactions.
Another idea would be to keep it in a terrarium where it can't escape and feed it like a pet until it's hatched out from it's cocoon. The adults are quite pretty and don't feed. All the eating these giant silk moths needed was done as a caterpillar. They don't drink water either, they get their moisture from eating fresh leaves. The adult moth only has about 2 weeks to do it's thing, mate, and if female, lay eggs before dying. Best to release once they hatch from their cocoon. In this way you can select which leaves it eats off your tree, perhaps from branches you were thinking of pruning this fall anyhow.