Eureka, I don't think he's fictive, I think I've found him and can link him to children.
What I had forgotten was that the Gillingham register for the period was in Latin. Hieronimus translates as Jerome (not Jeremy but surely this is the same man)
His birth won't be in the register because it doesn't start until 1565.
In 1565, Hieronimus Grene's daughter Alitia (Alice) was baptised. Hieronimus was the father of 7 more children, the final one being another Hieronimus, baptised in 1579.
Hieronimus Grene [senior] was buried in 1585.
In 1605, Hieronimus, the youngest son married a Maria Pyke. They had 7 children baptised between 1607 and 1622
The Dorset Archives has a series of four records relating to copyhold land, which were formerly part of the Chantry of St Katherine in Gillingham but post reformation belonged to Sherborne School who acted as Lords of the Manor (Nat Archives). These are in English.
The first from 1602 records that the property was formerly held by Jerome the father (deceased) together with two other lives, John Francis and his wife Rachel. The new lease was granted to Jerome the son, the term was the lives of John and Elizabeth Francis (children of John and Rachel. I don't know what it is but there may well be a close relationship between the Francis family and Jerome Grene. )
In 1622, a new lease was granted to Jerome who by now had several children of his own. No mention of the Francis children. The other two lives on this new lease were his elder sons Jerome and John. The final record from 1629 relates that Maria, Jerome's widow stated that her late husband had surrendered this property to a John Waye and Thomas Reeke before his death
If others agree, I'll, change his name to Jerome.(with Hieronimus as alternative and also Jeremy because otherwise we might get more duplicates ) ; He obviously used the English Jerome in his everyday life)
I've extracted and translated all the baptisms so can document the children and could create new profiles for them but apart from Jerome junior I don't know anything more about them .I haven't time to research further. There are an awful lot of Greenes in the Gillingham register. Someone may eventually link to them.