Trace the timeline of the disease to reach the bug. The story has two acts. The first act is a genital sore that does not hurt and feels firm; a painless indurated ulcer of this kind is the primary chancre of syphilis. The second act, arriving roughly two weeks to two months later, is a generalised maculopapular eruption that famously spreads to the palms and soles; this is the secondary stage of the same infection. Because both events belong to one disease, the organism behind them is the spirochaete Treponema pallidum. The remaining options each clash with the clinical detail: Chlamydia in its LGV form gives a fleeting painless ulcer but then painful lymph node buboes rather than a body rash, Klebsiella granulomatis produces the slowly spreading painless beefy ulcers of donovanosis with no secondary rash, and Haemophilus ducreyi causes chancroid, whose ulcer is painful and therefore contradicts the painless ulcer in the question. The painless ulcer plus later rash therefore fixes the diagnosis on syphilis.\[\boxed{\text{Treponema pallidum}}\]