I use warm water cause by the time I’m done, the water will cool off already. Then spam baby fever patches on the back of the neck, forehead and armpits. (I don’t wanna expose baby to too many medicines so I didn’t give panadol before 1. But also because my baby’s temp didn’t go beyond 38). After sponging (not too damp), I just let baby go topless or holey clothings, no blankets or swaddles so the heat won’t trap. Works for my baby.
Never tried lemon soap/bath but I saw mummies mentioned rubbing baby skin safe lemon essential oils on the back and chest during fever or flu is useful. You might want to look into it.