How could you describe the nature of Booker T. Washington's influence in helping, enlightening, and sustaining the narrator in Invisible Man?

In the first Chapter of Invisible Man, "The Battle Royal," the Invisible Man wants to be a young Booker T. Washington.  The speech he gives to the white members of the Southern town's social elite is filled with Washington's rhetoric:

We of the younger generation extol the wisdom of that great leader and educator.

But, he switches rhetorical modes and starts to sound like W. E. B. DuBois, a more outspoken Black activist of the time, an enemy to Washington.  He says: "Social euality" (DuBois) instead of "social responsibility" (Washington).

This rhetoric is too harsh for his white audience; they don't want to hear a black man demanding equality.  They expect him to be submissive and talk only of hard work, like Washington, not the political demands of DuBois.

To use a Malcolm X analogy: Washington is like the "House Negro": he is a moderate, acquiescent, advocating slow steps toward progress, playing by the rules.  DuBois is a "Field Negro": he hates his status and wants his freedom immediately and is willing to openly disobey white society.

So, the Invisible Man is caught between the two, caught up in the words of his grandfather:

I'd overcome them with yeses, undermine them with grins, I'd agree them to death and destruction. Yes, and I'd let them swallow me until they vomited or burst wide open.

His grandfather looked and sounded like a Washington, a house negro, but he was really a DuBois, a house negro:

Grandfather had been a quiet old man who never made any trouble, yet on his deathbed he had called himself a traitor and a spy, and he had spoken of his meekness as a dangerous activity. It became a constant puzzle which lay unanswered in the back of my mind.

So, the Invisible Man will spend most of the novel trying to find identity and freedom in the smalls steps advocated by Washington (work) and the bold moves advocated by DuBois (the Brotherhood).  Neither work: he is still invisible.  Unfulfilled, he finally rebels at the end and becomes an Invisible Man, waiting, hiding, hopeful to emerge one day as seen by himself and society.