Is it permissible to pray at home instead of the mosque for fear that the youth who see me as a role model might be led astray after I shaved my beard?

The questioner is not permitted to add the sin of not praying in the mosque to the sin of shaving his beard. His unwillingness for those who look up to him to see him is not a legitimate religious excuse for not praying in the mosque. He can pray in another mosque or use evasive answers if asked about shaving his beard, and he must take the initiative to grow it. He must rectify his heart and strive in that, for on the Day of Judgment, nothing will benefit except a sound heart. Let him beware of being among those whom Allah meant when He said: "Those are the ones whose hearts Allah does not intend to purify. For them in this world is disgrace, and for them in the Hereafter is a great punishment." His resolve to repent and his feeling of guilt are evidence of the sincerity of his faith.