Hi! You are right-- your "long" post is very helpful. I personally agree and think the more info and background, the more we forum folks can help you.
Sounds like you are doing things right between your new dog and your cat. I especially appreciate that you are keeping them safe at ALL times... and supervised. Now all you have to add is some good ole counter conditioning.
Oh, and personally I disagree with your trainers advice to wait for them to work it out themselves, even at a distance. Some animals acclimate on their own, many do not and certainly need our help to be safe and comfortable and not stressed. I have counter conditioned my very fearful cautious Gracie dog to our Tortie cat who looooooooves to stare at Gracie.
Man, cats love to mess with dogs, right?
In addition, I have actually taught my Gracie to call all of our cats and dogs to eat their meals. No one eats til Gracie dog calls each animal (specific bark) and tells me to put their food bowls down in a very specific order! Yup, Gracie dog feeds Tortie cat now. I keep saying I need to make a video of this because it is pretty dang amazing, esp if you knew how fearful Gracie was of this cat...and everything else in her world.
Used to be we were terrified Gracie would eat Tortie, or the reverse!! Seriously. Very scary between them.
I actually almost lost my eye during training years ago when I misjudged distance between them and picked up the cat when Gracie suddenly charged her ---and got back clawed near my eyeball!!! I call this "operator error"
or my fault!!
Live and learn. And counter condition!!!
And now.... I have done the same counter conditioning with my newer Puma pup and she helps feed the pets, too!
In fact, today it stormed here like crazy and so as part of my thunder training, I took Puma to several fun stores, including her fave pet shop... and there we visited the rescue cat Sophie that was living there in a large cage waiting to be adopted.
We practiced our counter conditioning with Sophie cat.
I asked Puma to sit or lay down in front of cat and I gave her lots of small yummy high value (bbq chicken) food bites for looking at the kitty. I ask her "Puma, where's the kitty?" and she knows to look at the cat, look back at me, and she gets the food.
It is our fun game that I play with my dogs. We also play "Where's the... squirrels, deer, cats, other dogs, skateboards, stollers, babies, birdies, you name it!!" This fun game works wonders to acclimate dogs to just about anything that may stress them.
In the beginning when it was new and a challenge for Puma to remain calm near kitty, she got "paid" (fed the treats) for just looking
at the cat. She was not required to do anything else. I just wanted to change her mindset about seeing the cat. Seeing cat=yummy treats, good things happen.
Over time I have shaped her behavior into doing advanced things such as please sit or lay down in the presence of the cat.
My gauge is always if my dog can perform her tricks for me in the presence of the cat (or other dog or stressful stimulus) than we are A-Ok.
If Puma got too hyper or barked, I just backed her up away from the cat and resumed the training/conditioning at a greater distance.
Today she did excellent and zero barking or excitement, very calm as she laid down next to kitty's cage and watched her peacefully.
We also practiced seeing the many cages of birds, ferrets, rabbits in the same exact manner. She did fantastic today, esp near the chirping moving lil birds which used to be her tough challenge. She went right up to their cage and looked at them with no reaction. Interest, but no reaction. Yes! Good job Puma!! Last week she was still a bit reactive and over excited to see the cage o lil birdies and we had to back- it- up!!
So today was DOUBLE BONUS training day: I got to counter condition Puma pup to crazy thunder storm AND a cat AND a bunny, ferrets, and birdies.
So much fun...but too much fun....I was late for work again....me bad. Practicing counter conditioning really, really works wonders for so many things with dogs!!!