Something like this: https://www.gardensafe.com/products/insecticide/bt-worm-and-caterpillar-killer.aspx

hopefully I can find this in a garden hose spray so I can spray the entire tree.
These things ate every leaf off my tree two years ago and were all over everything on my patio.

That doesn't mean it doesn't exist. It does come in a concentrate that you can mix with water in a sprayer.

It is safe to use and only kills certain insects. It does not kill pollinators and it won't hurt birds who eat the BT-treated caterpillars. It's used in organic gardening.

Trees and critters have been dealing with insects since before people walked the earth. Early leaves feed the bugs; later leaves are too tough or bitter.

Kill 1 predator with poison and hundreds or thousands of pests will then survive to make more pests.

and they ate every last leaf off my tree and my surrounding neighbors trees and the birds did not eat these caterpillars.
They poop worm poop on everything and crawl on everything.

A fruitless mulberry is a male. Only the female mulberry trees bear fruit. Pollinators must carry pollen from the male to the female flowers or there won't be any fruit.

Trees produce tempting leaves in spring, which encourages pollinators. After blooming season, they produce tough, bitter leaves.

Birds won't eradicate the caterpillars, and probably won't eat as fast as you'd like, but they do eat them. That worm poop is free fertilizer -- it's a lot of stuff the tree is able to pull from deep in the ground that other plants can't reach.

Poison the caterpillars and you reduce the local songbird population as well as deplete the soil. That's without even considering the direct effects of the poison.