With all due respect, I do not think you are depriving Max of anything if you don't bother with the keto diet.
https://sciencebasedmedicine.org/ketoge ... s-science/
I used to have a 'live and let live' attitude about claims like that, and I don't know when I lost all my tolerance -- maybe when donald trump made it into the White House. The stakes are just too high to let rank stupidity go unchallenged. And so I will express my opinion of the keto diet: hogwash. The mere fact that it is promoted by Joe Mercola is an assurance that it is the most egregious bullshit scam that it can possibly be. And Ralph Moss, too?! OMG, peach pits cure cancer! This is all the shite I waded through in 1999!
FWIW, for the 7+ years that Sander lived with cancer, I did NOT feed him a keto diet or any approximation of one. He ate oatmeal; he ate barley; he ate quinoa; he ate amaranth. He didn't eat corn or wheat, but IMO no dog should. He ate plenty of vegetables, which are (gasp!) carbohydrates. He ate fruits when he wanted them; he seemed not to want them very often, after a while.
Your idea to give Max more veggies is probably a good one; I sympathize with the difficulty in keeping weight on a senior dog who has the health issues that Max is facing. Good luck. Don't give the keto nonsense another thought.