Is it permissible for a girl to invite a non-Muslim man to Islam, and would it have been better to use another method to avoid misunderstanding?

There is no legal impediment for a woman to invite a non-mahram man to Islam, provided that Islamic guidelines are observed. It is preferable to seek the assistance of knowledgeable and wise men if possible. The method of Dawah (inviting to Islam) employed is good, and there is no harm in praising the good qualities of the person being invited to win his heart, as the intention is what matters. Caution must be exercised to prevent the person being invited from exploiting this.