Given the clearer perspective we now have and given the fact that Chesterton himself was plagued by depression*, it's not about any display of exuberant behaviour or the flaunting of good health that he is referring to but rather the state of exuberant health itself, which in his view, is both defiling and offensive. In which case anep's rendering at #3 "απαλλαγμένη από κάθε οχληρή πληθωρικότητα υγείας" comes closest to being correct. Perhaps it could do with some minor tweaking.You gave the right suggestion #2 then - and you should have said something much earlier.
.... and where "sickly amount" = "sickening amount" (="nauseating amount").She's not defiled with/tainted by a "sickly amount" of good health = She's defiled with no "sickly amount" of good health.
(where "sickly amount" = " offensive exuberance")