Why are you making it narrower? It will tip easier. I have put a dog crate on a kid trailer frame. It handled OK. But test it on a long ride before the trip, it will slow you down a lot. I got rid of mine. In August the heat coming up from the pavement is very important. Having a roof makes it worse.
I have the trike in my Avatar, the basket is higher and cooler even without shade. Also note that your bike and you block most of the air getting to the trailer. The dog will over heat quickly, compared to you. After a few thousand miles with the dog in the trike, and a few hundred in the trailer, using thermometers in different locations in summer, I would never take a dog on a bike trip. Use the dog cooling vests you can find on line.
Also note, a black dog gets hot much faster than a white dog. I don't ride the dog in the trike in anything over 70 degrees at the start of the ride now.
Dogs have a higher normal temperature than we do, and long fur could be similar to wearing a sweater.

Having the trailer slightly to the left gets the right hand wheel of the trailer further away from the curb, or the edge of the pavement. A good thing.

Actually before you put the crate on, try towing the frame around in traffic and on rural roads. You might not like it.