How can one reconcile performing the Sunnah prayer of Fajr at home, while being keen to be in the first row, and not offering many prayers before Fajr other than the regular Sunnah prayer?

What you mentioned about the obligation of striving for the first row is incorrect. Rather, it is something encouraged in Sharia, not obligatory. One does not incur sin for not striving for it or for not praying in it. It is recommended for one who has prayed the Fajr sunnah at home and comes to the mosque before the Iqamah to pray Tahiyyat al-Masjid (greeting of the mosque). Therefore, be diligent in performing the Fajr sunnah at home, then go early to the mosque to catch the first row, and pray Tahiyyat al-Masjid immediately upon entering.