Club that chap joined to go swimming (5)
I believe the answer is:
Here is my best explanation:
'go swimming' is the definition.
(I've seen this before)
'club that chap joined' is the subsidiary indication.
'club' becomes 'bat' (bat is a kind of club**).
'that chap' becomes 'he'.
'joined' means one lot of letters go next to another.
'to' is the link.
(Other definitions for bathe that I've seen before include "Bask", "Swim in sea", "Wash in water", "Immerse oneself in water", "Swim or paddle", "Cleanse the body", "Take dip in sea", "Suffuse", "Wash by immersion" and "Swim for pleasure, and clean oneself".)