Our little monster is the same about cooked chicken, if you have it you have obviously made a mistake, that is his food and while he is a gentle god he can also be a jealous god.
As for the fat, we have recently tried using fat from pork roasts, the fat cap is thick and soft so we can take a little of that and add it to his containers. I tried more refined fats, he will definitely eat butter, but he ends up missing the stop point and having loose stools if we use a refined fat.
This is the exact type of situation for BARF or PMR. BARF (Biologically Appropriate Raw Food) and PMR (Prey Model Raw) basically focus on hitting the nutritional needs of animals like cats and dogs by matching what they would get from prey animals. This includes a fair bit of digest for dogs but less so for cats.
My cat is doing very well on BARF and is not showing his age at all. He used to have bad hairballs, lots of itching, and very bad fur. He is now sleek and healthy with very shiny teeth and lots of energy.
His food consists of chicken cutlet (for skin, bone, and meat), chicken or lamb liver, chicken or lamb heart (for taurine), sheep brain, usually lamb kidney but sometimes beef, and then some selection of rotating meat which includes beef, pork, lamb, chicken, salmon, and prawns (sometimes called shrimp). It is about 7.5g for each of the heart, liver, kidney, and brain, then the bone is varied to change stool consistency (more bone makes harder stool) so he has an easy time there, then the meat is rotated to keep deficiencies in any specific meat at bay.
It takes about half an hour once a fortnight to make his food and he is fed every morning. He sometimes carries his meat somewhere else to eat it so we have to clean carpet or surfaces, but honestly, he is just way too cute for us to try and do something about it.
Anyway, maybe switching would help, the process of switching takes a while and requires building jaw strength so it is a little slow going at first, but the improvement to health is pretty great and it isn't that much effort. Oh, and his poop smells way less, and there is less of it. He has much smaller and smoother poops, nothing runny, nothing girthy, and the cat litter is able to contain the smell more effectively than previously.
My cat only eats grass when he wants to vomit. He will clean himself, get too much hair in his stomach, then yowl to go outside. He then eats grass, vomits, and is chill. When I am more on top of brushing him this happens less, but if I neglect his brushies he needs to purge the hair.
Lol, my cat eats a diet of chicken, prawns, some beef, and organs like liver, heart, kidney, and brain. He is extremely healthy and does not look his age (13). He started on kibble but we transitioned him to meat because of hairball and general health issues. His teeth are clean and strong, his coat is shiny, and his eyes are clear.
Honestly, cats are predators. They eat meat. Feed a cat mice and you are close to what mine eats. It would be really strange if they were ok eating rice, corn, and brocoli.
Some cats are just like that. Mine will look at a laser dot and watch it for a while, but he really needs to actually catch something to be happy with it and will lose interest without winning.
I got some toys from Temu the other day and he loves them. They are balls that roll themselves and have a little frame with a tail. The movement is somewhat chaotic with pausing, shaking, and reversing. Because they are balls they tend to drift to the left or right so they don't just go back and forth in the same line. He will watch them move until they stop then have a big pupil dilation reaction watching them paused and eventually hit them. If they move before he bops them then he keeps watching, but if he acts quickly enough he bops it and it looks like it reacts and runs away.
He also absolutely loves the cheap versions of Da Bird, a carbon rod with a toy on the end of a line. It looks like a bird when you swing it around the room and you can make it land on things and wiggle which really gets him engaged.