Taking your cat to the vet for a workover to see if there is any medical problem would be a good thing to do. Some obesity is due to a medical condition, but of course, blood work would have to be done to know. And, sound advice on how to help your cat lose weight would be good to hear from a vet. It is dangerous for cats to lose weight quickly.
I don't think shaving cats is a good thing to do in general. They did evolve in a desert climate, with fur, after all. Fur acts as an insulator from cold AND heat. Cats shed more when it's warm, to regulate things naturally. Perhaps some recently created long-haired breeds have more fur than "nature" would have supplied, and could do with some (not all) of their fur shaved in very hot weather. And, there are some cats with triple layer coats who might benefit from at least a crew cut of sorts, but not a total fur removal. Some long-haired cats get very tangled mats, that do need to be cut. But I would not shave any fur off of an ordinary short-haired cat.
My cats often go outside to lounge on our latticed-in porch when we have the AC on. They seem to prefer things on the warm side (though not terrible heat in direct sunlight, of course.) They will be snoozing out there in the 80 degree heat, quite happy, while having the choice to come into the climated controlled 70 degree house. Probably because their body temperatures are higher than that of humans, so they feel the cold faster.
You will need to help your cat with his grooming, since he is too fat to reach his back. Most of my five cats love for me to comb them with a flea comb. It gets out all the loose hair and any debris much better than a brush, and I think it feels to them like getting a nice back scratch (just go easy on the spine or it hurts them.) Then wipe wipe your cat down daily with a washcloth wet with warm, not hot or cold, water to be a substitite cat tongue, to get the dust and excess oils off. That can turn into a mighty nice bonding, lovey time.