I checked the Library index for St James Church and found this extract:
"The church was not assigned a Parish until 1844". This probably accounts for his burial here, even though he seems to have lived in Town (and married at St Nicholas 1768).
The first naming of Penny Lane, found so far, is the 1840s well after his death ((1799), or that of his eldest son (also James Penny d.1820). The track existed for many decades prior to this date and likely already had a name. As the census was 1841 I'd assume it was Penny Lane prior to this decade.
A trawl of early newspapers would seem to be the most likely place to find street-name changes and announcements about creating new streets etc. I'd hate to be the one searching for the word 'penny', though.