Is it established that the Prophet (peace and blessings be upon him) used to bathe between the two 'Isha prayers in Ramadan to strengthen and invigorate the body?

There is no authentic evidence from the words or actions of the Prophet, peace and blessings be upon him, that proves the legitimacy of performing ghusl (definition) before 'Isha in the last ten nights of Ramadan (definition). The narrations that mention this are weak or disconnected.

However, for one who performs ghusl with the intention of being energetic for night prayers, or for perfuming and adorning oneself, his action is good. Similar to this is what has been narrated from some of the Salaf regarding performing ghusl and adorning themselves on the nights when Laylat al-Qadr is hoped for.