Doubt you will find a baptism as parish records don't go back that far (Aisthorpe where the family lived starts in 1595)
According to this Pedigree his father died in 1506 so as he had several younger siblings, he was born some years before that If you could find an inquisition post mortem, for the father, it might tell you how old his elder brother was on his fathers death . Elder brother Alexander was old enough to have an heir, John who was 47 years 6 months in 1556/7. (so born c 1509)
Whilst looking for an IPM on the National archives site, I came across two intriguing looking records held by the Kent Archives and concerning a large bequest to the town.
"Copy of the will, endorsed 'thys ys the forgyd wyll of Henry Hatche, whereuppon Parker and Goldfynch and the seid Hatches wydowe being the Ladye Amcottes dyd stande, to dysprove the true wyll of the same Hatche provyd at Canterburie and ther remayning of recorde' (1533)"
"Extracts of the will, with notes that Hatch died in 27 Henry 8, and extract of incorporation of the town with the right to hold property in mortmain [i.e. the Hatch estate], and with notes that Lady Amcottes died about Michaelmas 1573"
From elsewhere, I found that Henry Hatch was a merchant adventurer and wholesale fishmonger, died of the plague in 1533 and left a fortune to the town of Faversham. His widow contested the will and it wasn't settled until after her death There is a brass in the church there that commemorates both Henry Hatch and Sir Henry Amcotts, L.M. of London, Master of the Fishmongers' Company.
Does this suggest that Henry Hatche's widow became Sir Henry's second wife? According to the pedigree she died in 1573. I don't know but an interesting 'rabbit hole'