Is it permissible for the aforementioned young man to lead the prayers in the mosque attended by the father of the girl he wishes to propose to, in order for her father to get to know him and facilitate his proposal, or would that be considered showing off?

If your original intention in going to the mosque was to draw closer to Allah and perform the prayer, and the matter of marriage was secondary to that, then this does not harm sincerity nor does it lead to showing off. It is excusable for the self to have an immediate worldly benefit if it is secondary to worship and not an original intention. It may be better to seek help from some virtuous individuals to speak with the girl's father, and to increase in supplication. You should also inquire about the girl, and perform Istikhara after approval.