My concern with narrowing the trailer would be stability in turns. We ended up crashing our tandem a couple of times when I took a turn too fast when pulling the dog trailer, (1st generation burley kid trailer) and it tipped over. The dog(s) have a high CG, and get thrown to the outside of the turn, so stability is much worse than with non-biological cargo, or a seated & belted kid. Fortunately these were sharp turns, so "too-fast" wasn't all that fast, and no harm done except maybe a skinned elbow or bruised hip on the humans. I may have clipped a curb with the inner trailer wheel one of the times...which would be an argument for narrower tread, but I'd opt for wider and more stability regardless.

Training dogs to ride in trailer: If the trailer is open, Use a harness, and rig a short lead so you can clip them into the trailer. If the front of the trailer is open, you need this in case you have to stop quickly anyway.
Start out with very short rides to someplace doggy fun...like a park or pet store. Bring treats for when you get there. After a few trips like that, our dogs were jumping in the trailer and barking at us to "go already!"

I no longer have big dogs, so the little guy I have now rides in a front basket now, and he loves that.