Bard's deep thinking about the situation is an incredible read.
Yes, he comes from a society where the thought of a woman fighting is almost offensive, in his case it doesn't come from a place of spite, just education, and even consideration, but even if gets him worked up as a soldier that dealt with life and death... he knows that beneath her naivety and inexperience is true sincerity.
He recognizes too that as a person, she can desire to fight for what she believes in, and that... he himself shares that desire to follow his heart and conviction.