Interesting speculation. We do have the author's word that hobbits are subject to illness. See this passage from the Grey Havens chapter where Tolkien describes the wonderful period of peace and prosperity that followed the War of the Ring. The italics are mine...
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Altogether 1420 in the Shire was a marvellous year. Not only was there wonderful sunshine and delicious rain, in due times and perfect measure, but there seemed something more: an air of richness and growth, and a gleam of a beauty beyond that of mortal summers that flicker and pass upon this Middle-earth. All the children born or begotten in that year, and there were many, were fair to see and strong, and most of them had a rich golden hair that had before been rare among hobbits. The fruit was so plentiful that young hobbits very nearly bathed in strawberries and cream; and later they sat on the lawns under the plum-trees and ate, until they had made piles of stones like small pyramids or the heaped skulls of a conqueror, and then they moved on. And no one was ill, and everyone was pleased, except those who had to mow the grass.
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If Tolkien thought it was important to mention there was no illness in this era, we can assume that there must normally have been illness. It's true that hobbits lived extraordinarily long lives. We see no explicit mention, for example, of infant death. Yet the Brandybuck geneology does list two names with no descendents and no dates of birth of death given. And there are a number of names in the Samwise tree that show a birthdate but no descendents and no death date. Some of these are Sam's children but some appear much earlier in the family tree so they wouldn't have been alive when the Red Book was written. Did all of these folk marry and live long lives? We can't be sure since Tolkien gives us information on only a few of these individuals. It's possible that some fell prey to illness. And, even within the chronology, some did die at a "younger" age....witness Bungo Baggins who expired at 80 leaving Bilbo an "orphan" when he was 36. Illness could have played a part in such deaths but simply isn't mentioned.