A Word To Use To Describe A Person's Non-Physical Presence

A lot of words related to mind and soul can be used here, like:

psychic or psychical: relating to the soul or mind.

emotional: relating to a person's emotions.

spiritual: not concerned with material values or pursuits.

inner: mental or spiritual.

cognitive: relating to cognition.

psychological: related to the mental and emotional state of a person.

intellectual: relating to the intellect.

mental or mindly: relating to the mind.

psychogenic: having a psychological origin or cause rather than a physical one.

Hope, it helped :)

Consider:

Dominating the room with his physical and cerebral presence...

It is a near synonym of intellectual:

2a : appealing to intellectual appreciation
    cerebral drama

2b : primarily intellectual in nature
    a cerebral society
    books for cerebral readers

_{Merriam-Webster}

"Persona" could be used that way. It's meaning is, more or less, confined to the projected character, which according to Cambridge online is often a distinct entity from the real character. However, since your question seems to point more to the perceived image than the real character that is hidden inside, it is a fitting word. The projected image of strength could very well reflect an interior reality.

persona noun [ C ]

the particular type of character that a person seems to have and that is often different from their real or private character: "He had a shy, retiring side to his personality that was completely at odds with his public persona." (dictionary.cambridge.org)

And here is an example usage in a headline:

A dominant persona and its many downsides: Those in a leadership position can easily end up putting off people than winning them over. (gulfnews.com)

Your example sentence [fragment] asks for an adjective, but using persona it could be rephrased:

"Dominating the room was his persona..."