Which is more befitting upon entering the mosque shortly before the iqamah: the regular Sunnah prayer of Dhuhr – for example – or the 'Tahiyyat al-Masjid' (two rak'ahs upon entering the mosque)?

It is sunnah for whoever enters the mosque not to sit until they pray two rak'ahs. If they sit before praying, it is sunnah for them to stand up and pray two rak'ahs. If they enter the mosque and pray the regular sunnah prayer, it suffices for the mosque greeting (tahiyyat al-masjid), because the purpose is to pray two rak'ahs before sitting. It is not a condition to intend the two rak'ahs as a greeting; rather, if one prays two rak'ahs with the intention of prayer in general, or a regular or non-regular supererogatory prayer, or an obligatory prayer—whether performed on time, made up, or vowed—that suffices, and the mosque greeting is implicitly achieved.