I have my bettas in the room without heaters in the summer time . And i have thermometers in each tank and the temperature in the tanks the same as the room.....I love the thermometer strips that you can stick outside of the tank. If you see one i do recommend them. It sensitive to the temperature changes.
The thing is that no matter how experience people are you always can get sick fish form the store. I bought a lot of bettas . I have 8 and i bought about 7 for other people. You always have chance to get sick one.
I saw someone recommended to have emergency kit at home. I always have aquarium , epsom salt and medications at home in case i get sick fish.
I hope you fish will be better. If you think he is lethargic i would treat him . Just watch him and see his behavior and give us update so we can decide what to do . And i guess other people on the forum will write what they think.
Any symptoms on his body? Any spots, bloating, cottony white patches....