I am trying to understand the sentence "Tom loves Jane with his whole being" without involving the word "with".
Does the sentence "Tom loves Jane with his whole being" have the exact same meaning as "Tom's whole being loves Jane"?
with his whole being is a metaphor meaning completely and without reservations.
It is similar in meaning to wholeheartedly, which means with (someone's) whole heart. The heart is a metaphor for the emotions, so wholeheartedly means with all of one's emotions.
being is what defines a person- their essence, personality, spirit, soul: the part that goes away when somebody dies. with his whole being means with every part of his personality.
The following sentence might be a suitable way of explaining your original sentence:
Everything that Tom thinks, says or does reflects his love for Jane.
The phrase "Tom loves Jane with his whole being" is slightly idiomatic - a bit like, for example, "Tom loves Jane with his heart". You wouldn't really say, "Tom's heart loves Jane".
"...with his whole being" is suggesting complete or unfaltering love.