Gymglish defines in the spirit of as
in the attitude of, in the essence of (idiom)
Ex: It's not in the spirit of teamwork to betray one another. (That's not the essence of working together at all.)
Whereas WordHippo says that it means
In keeping with given expectations, norms, ideals, or rules.
So I think that the phrase In the spirit of full disclosure means wishing to be completely open/honest, (just like a "full disclosure" is in law or business).
Note: full disclosure is a formal set phrase meaning :
the disclosure of all relevant information, without the withholding of significant facts which may bias a decision.
One can clearly discern irony in this sentence, as the intentionally formal (or even pompous?) register of the phrase is used to increase the hilariousness of the situation.