I highly doubt it's because he's not getting enough protein. Not eating enough is more likely the answer.
Max has a liver condition from what you have posted and he really needs to be on low protein food with NO red meat including lamb, venison, and beef. That's why he's not eating right...Diamond food has those ingredients.
Liver compromised dogs don't like to eat because food makes them feel sick...That's why no red meat is so important. My dog has been on low, nonmeat protein for 3 years and he's healthy as any dog
I really think he's acting this way because 1)he's not eating enough and 2) his liver problems
I'll be praying for your little guy. I know you're extremely concerned about him...Hopefully you can find a specialist for Max.
Yes, karo syrup or honey or sugar water is okay to give to him to get his sugar levels up.
Good luck!!