At that distance, sometimes there is only guesswork if there are no records other than parish BMDs. Not even father's name was recorded for marriages, and mother's name was not usually recorded for baptisms.
FamilySearch shows a marriage that would be at just the right time: Oct 1679
https://familysearch.org/pal:/MM9.1.1/NKGM-2HSEdward the III would have been 22, and his son Joseph/Josias then born between 1679 and 1685ish to marry in 1707 (oops, not 1807 as I originally typed).
That marriage was to Eliza Sutton at Saint James Dukes Place, London.
Searching within that batch, M02248-2, there were two other Bellamy marriages at the same place around the same time:
Thomas to Elizabeth Cesar, 1680
John to Mary Golding, 1680
There are maps showing the two parishes here:
St Magnus the Martyr
https://familysearch.org/learn/wiki/en/St_Magnus_the_Martyr_with_St_Margaret_New_Fish_Street_and_St_Michael_Crooked_LaneSt James Duke Place
https://familysearch.org/learn/wiki/en/St_James%27_Duke%27s_Place#1831_description(StJDP is the little unlabelled bit of parish in the upper right corner of the map showing StMtheM in the lower left)
The IGI has these batches for marriages in StJDP (no baptisms or burials):
M022481 1700-1868
M022482 1664-1682
M022483 1682-1687
M022484 1688-1691
M022485 1692-1700
It has these batches for marriages in StMtheMandStMNFS (no baptisms before 1800, no burials):
M079041 1675-1754
but the only marriage it shows for the name Bellamy is
Elizabeth Belamy to Samuel Sanklen 1676
Since that is the period Edward III born 1657 would have married, maybe the marriage in StJDP is worth considering!
Oops, we're crossposting -- I know that was you, and it was me too.
Threads about the same family, wise to cross ref if starting a new thread.
PS -- if you've made the connection to Adm George Somers (who had no issue??), do you want to post that back in the original thread?
I just got a little invested in it and would be very curious to see the outcome!