Talk to Pediatrician on Baby Sucking Fingers

My baby is 4 months old and exclusively formula fed ( Meiji FMT). He has developed a habit of sucking his fingers from last 1.5 month almost. Initially we thought he is perhaps teething which is why he does it but with the passage of time his habit turned into a routine thing. He puts his hands in his mouth all the time even while asleep. Twice he got infection and got treated for that. He is not drinking milk and not gaining any weight. We have tried every possible thing to stop him like putting on mittens, using musabbar, by taking his hands out over and over, by politely telling him not to do so but he gets very cranky and fussy if we stop him from doing so. He is getting weaker day by day. We are very upset pls help us how to stop him from sucking on his hands and how to increase his milk intake. He was born a healthy baby of 8 pounds but now he is underweight which is very depressing. Pls help us and let us know heart to do in this situation.