I'm afraid there aren't any registers of individuals in the parish workhouse of St Dunstan in the West at that date.
There is a Workhouse Committee Minute book, 1799-1809, at the London Archives (P69/DUN2/B/065/MS12016), but I believe it contains chiefly administrative records. For further information on its contents you could ask the archive:
https://tla.libanswers.com/form(St Andrew Undershaft is a different and unrelated parish, some distance away.)