Everything in moderation. While fat is necessary in their diet too much will do a great deal of harm to their internal organs, plus it will make them have the runs worse than usual which may lead to dirty bums and irritated skin that wouldn't be suitable on a plucked bird.
What's the point in fattening up a chick that isn't even at butchering size? The way I see it is they need to grow their bones and then pack on the muscle and eventually the fat. Protein is what you want if you want plump meaty birds, much like body builders bulking up on protein intake so should the fast growing meat birds. That being said, more is not always better. They should be getting a proper meat bird feed or at least a good chick starter to give them all they need. Too much of this or that creates an unbalanced diet which leads to health problems. The main issues with meat birds is organ failure (heart congestion and/or ruptured fatty liver) and skeletal issues, particularly in the legs. If they grow too fast their bones cannot support their weight, same story with their hearts. You'll hear it all the time how meat birds will just drop dead from heart attacks or internal bleeding from a ruptured liver and it becomes more common the larger the birds get.
There are tried and true methods for raising healthy meat birds, that's why there is specific meat bird feed and the recommended 12hrs on and 12hrs off feed. I'm not saying other ways won't work but adding a bunch of fat to their diet isn't it. If anything I'd maybe look into the French way of finishing bresse chickens. You feed them a certain diet, typically grains soaked in milk, a week or more before you plan to process them.
https://www.ambresse.com/finishing-american-bresse.html
If you want big birds then just raise them longer than 8 weeks. Many won't live too much longer after 2 months if kept on a meat bird diet as they will just keep growing. If you limit their feed and give them something less packed with protein they can live a somewhat normal, healthy chicken life.
I did a small batch of 10 Cornish cross this summer. Processed 7 of them at 7.5 weeks old for ~7lb carcasses. Butchered 2 more at 9 weeks for an 11lb and 8lb carcass. The last one I kept till I hatched one of her eggs, she was about 25 weeks and weighed 11.5lbs dressed. I'd say about 10-12 weeks would give you the largest size but still be tender and relatively healthy.