The Dictionary of Irish Biography
https://www.dib.ie/biography/aungier-sir-francis-10275 and The Consolidated Index to the Records of the Genealogical Office, Dublin, Ireland Chapters A-B-C
https://www.nli.ie/sites/default/files/2022-10/mcanlis-consolidated-index_vol.-1_a-c.pdf list Sir Francis Aungier with first wife of the name Douglas Fitzgerald, a daughter of the fourteenth earl of Kildare - they married in the 1590's.
So the profile of Lettice Aungier has the correct parents.
Looking through the East Clandon register, there are the following baptisms/burials
Baptisms
1592 02 November Frauncys Augeir daughter of Frauncys Augeir Esquire
1594/5 23 March Garret Augeir son of Francis Augeir Esquire - 'his godmother was ye Lady ?Woolly and his godfather Mr Thomas Garrett ....'
1596 12 September Ambrose Aungeir son of Frauncys Aungeir Esquire
1597 27 November Thomas Aungeir son of Frauncis Aungeir Esquire
Burials
1597 17 December Richard Aungeir Esquire buried in St Thomas Chancel
1600 01 June 'Douglas Aungeir the wyffe of Frauncis Aungeir Esquire was buryed the first day of June'
Also, the Will of 'Sir Francis Aungier, Baron of Longford, Lord Aungier, Master of the Rolls in Ireland' is available on the National Archives website here
https://discovery.nationalarchives.gov.uk/details/r/D865942
Edit to add details of baptism/burials and Will