Gavin didn't summon Lei.
Although the Sun Knight promised her that he would notify her once he discovered a clue, the truth was, after spending a few days together, he had already recognized Lei's true nature.
She was but a child.
Reality was not a fairytale. Boys and girls couldn't experience a true magical adventure because reality was just that - reality. Atoms constantly change according to basic law, and the beautiful fantasies of mankind did not reconcile with the truths of the universe.
Despite her many remarkable qualities and a future Gavin felt had potential, which was why Bedwell had chosen her as his successor.
However, even Bedwell hadn't foreseen his premature demise in battle. Initially, he hadn't expected Lei to take on any significant role under such circumstances.