Is a prayer of gratitude permissible instead of the salutation of the mosque upon entering it?

There is no specific 'prayer of thanks' (salat al-shukr) in Islamic law. The purpose of the 'greeting of the mosque' (tahiyyat al-masjid) is to perform two rak'ahs before sitting down. Any prescribed prayer, such as a regular sunnah prayer (ratibah), a prayer of seeking guidance (istikharah), or a prayer after ablution (sunnat al-wudu'), suffices for the greeting of the mosque.