Thanks for the pictures but it is still tough to tell - the top down picture lying down will not be accurate given that the body spreads, so ideally if you can get one while he is standing - top down....that is ideal.
Keep in mind that a Berner continues to fill out in his third and fourth year. The chest drops and widens - the dog essentially fills out.
Testing for weight should be done by feeling the rib cage along the side of the dog - to know what to feel for, open your hand with your palm facing downward and feel your knuckles ....this is what a dog should feel like and you should be able to feel his tailbone as well.
Another good guage is watching his weight...what does he weigh right now and is this above his normal or what he should be? You can take him in to most vet offices and just use their scale.....
If you are now feeding a diet food essentially, you should monitor his weight to see if this is doing anything.
Authority is not a well reviewed food or classified as "high end" given its high content of corn, corn meal and beet pulp and for this reason I do not have any info on it - such as calories per cup etc.
But I will assume it is low calorie and at only 3 cups per day, you are likely feeding far less than he requires so this should be resulting in some weight loss if his thyroid is fine etc.
Keep in mind grains stick on a dog worse than on us humans so if you think about what eating a lot of breads, pastas, and corn does to us.....double it for dogs. I find it difficult to reconcile low fat and diet foods when they still put so much grain in them.
Ok...lets look at what else you are feeding:
You did not specify what you are doing with the carrots but presuming they are raw and shredded/ground, along with cooked brown rice and 1/2 cup of raw ground chicken - this is approximately 275 calories with 35% of these calories coming from carbs, 33% coming from fat and the rest protein
Ok, as for sweet potato, peas and hamburger....
You did not say what fat level of ground beef you are using...so I will presume 15% fat and you did not specify how your serving the sweet potato or peas (these should always be cooked or will not be digested) - but this quantity is 400 calories with 39% of these calories coming from fat.
Just using these two examples, you can see the disparate level of fat and calories here. The second meal here at 39% fat will not be highly valuable in a diet aimed at reducing weight, depending on how many calories you are feeding in kibble.
My recommendation to you is to weigh him, and feed him one food twice per day to see how he does. Pick a good quality kibble, OR commit to a raw diet as mixing these is not providing a good balance of nutrients and may be the source of the issue.
I trust you bought Authority at PetSmart as this is their food brand and so of course they recommended it
I recommend if you go with kibble, you select a high end brand with more bio-available nutrients such as Taste of the Wild, Natures Variety, Orijen, Innova or Wellness. None of these can be found at Petsmart but all can be found in a good animal supply store. Look them up online to find a store near you.
Hope that helps!
Rachel